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Demonstration of the

On-Line Encyclopedia of Integer Sequences® (OEIS®)

(Page 12)

Fractions, Arrays, Real Numbers, etc.

The OEIS contains much more than just sequences of whole numbers. There are also sequences of fractions or rational numbers, triangles or other two-dimensional arrays of numbers, decimal expansions and continued fraction expansions of important constants, among other things.

In this page we shall give several examples of such sequences.

Rational Numbers or Fractions

  • Rational numbers (or fractions) are given as a pair of sequences, listing the numerators and denominators separately.
  • This is indicated by the keyword "frac", and the two sequences are cross-referenced to each other.
  • The Bernoulli numbers are a good example. This is probably the most important sequence of fractions in number theory. It begins:

    Sum
  • The numerators and denominators are listed separately as sequences A027641 and A027642 as shown here.
  • Note that there is a separate file http://oeis.org/wiki/Index_to_fractions_in_OEIS that gives the A-numbers of the numerator and denominator sequences. for a number of important sequences of rational numbers.

First the numerator sequence:

A027641 Numerator of Bernoulli number B_n. 73
1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051 (list; graph; refs; listen; history; edit; internal format)
OFFSET

0,11

COMMENTS

a(n)/A027642(n) (Bernoulli numbers) provide the a-sequence for the Sheffer matrix A094816 (coefficients of orthogonal Poisson-Charlier polynomials). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The corresponding z-sequence is given by the rationals A130189(n)/A130190(n).

Harvey (2008) describes an algorithm for computing Bernoulli numbers. Using a parallel implementation, he computes B(k) for k = 10^8, a new record. His method is to compute B(k) modulo p for many small primes p and then reconstruct B(k) via the Chinese Remainder Theorem. The time complexity is O(k^2 log(k)^(2+epsilon)), matching that of existing algorithms that exploit the relationship between B(k) and zeta(k). An implementation of the new algorithm is significantly faster than the implementations of the zeta-function method in PARI/GP and Mathematica. The algorithm is especially well-suited to parallelisation. Some values, such as B(10^8) may be downloaded from his web site. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 09 2008

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.

H. Bergmann, Eine explizite Darstellung der Bernoullischen Zahlen, Math. Nach. 34 (1967), 377-378. Math Rev 36#4030.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.

H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974; see p. 11.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.1.

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.

L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.

H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons

K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

K. Dilcher, A Bibliography of Bernoulli Numbers (Alphabetically Indexed Authorwise)

David Harvey, A multimodular algorithm for computing Bernoulli numbers, July 8, 2008.

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Peter Luschny, Die Riemannsche Funktionalgleichung als Grundlage der Bernoulli und Euler Funktion. (2004) [From Peter Luschny (peter(AT)luschny.de), May 02 2009]

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.

S. Plouffe, The First 498 Bernoulli numbers [Project Gutenberg Etext]

E. Sandifer, How Euler Did It, Bernoulli numbers

Eric Weisstein's World of Mathematics, More information.

Wolfram Research, Generating functions of B_n & B_2n

Index entries for sequences related to Bernoulli numbers.

Index entries for "core" sequences

FORMULA

E.g.f: x/(e^x - 1). Recurrence: B^n = (1+B)^n, n >= 2 (interpreting B^j as B_j).

B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).

Sum_{i=1..n-1} i^k = ((n+B)^(k+1)-B^(k+1))/(k+1) (interpreting B^j as B_j).

B_{n-1} = - Sum_{r=1..n} (-1)^r binomial(n, r) r^(-1) Sum_{k=1..r} k^(n-1). More concisely, B_n = 1 - (1-C)^(n+1), where C^r is replaced by the arithmetic mean of the first r n-th powers of natural numbers in the expansion of the right-hand side. [Bergmann]

Sum_{i=1..inf} 1/i^(2k) = zeta(2k) = (2*Pi)^(2k)*|B_{2k}|/(2*(2k)!).

B_{2n} = (-1)^(m-1)/2^(2m+1) * Integral{-inf..inf, [d^(m-1)/dx^(m-1) sech(x)^2 ]^2 dx} (see Grosset/Veselov).

Let B(s,z) = -2^(1-s)(I/Pi)^s s! PolyLog(s,Exp(-2IPi/z)). Then B(2n,1) = B_{2n} for n >= 1. Similarly the numbers B(2n+1,1) which might be called Co-Bernoulli numbers can be considered and it is remarkable that already Leonhard Euler in 1755 calculated B(3,1) and B(5,1) (Opera Omnia, Ser. 1, Vol. 10, p. 351). (Cf. the Luschny reference for a discussion.) [From Peter Luschny (peter(AT)luschny.de), May 02 2009]

EXAMPLE

B_n sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ...

MAPLE

B := proc(n) sum( (-1)^'m'*'m'!*combinat[stirling2](n, 'm')/('m'+1), 'm'=0..n); end;

B := proc(n) numtheory[bernoulli](n); end;

with(numtheory):seq(numer(bernoulli(n)) , n=0..40); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 08 2009]

MATHEMATICA

Table[ Numerator[ BernoulliB[ n]], {n, 0, 40}] (from Robert G. Wilson v Oct 11 2004)

Numerator[ Range[0, 40]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 40}], x]]

PROG

(PARI) a(n)=if(n<0, 0, numerator(bernfrac(n)))

CROSSREFS

This is the main entry for the Bernoulli numbers and has all the references, links and formulae. Sequences A027642 (the denominators of B_n) and A000367/A002445 = B_{2n} are also important!

Cf. A027642, A000146, A000367, A002445, A002882, A003245, A127187, A127188.

Sequence in context: A036946 * A164555 A176327 A129205 A098173 A180977

Adjacent sequences:  A027638 A027639 A027640 * A027642 A027643 A027644

KEYWORD

sign,frac,nice,changed

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com).

Then the denominators

A027642 Denominator of Bernoulli number B_n. +0
26
1, 2, 6, 1, 30, 1, 42, 1, 30, 1, 66, 1, 2730, 1, 6, 1, 510, 1, 798, 1, 330, 1, 138, 1, 2730, 1, 6, 1, 870, 1, 14322, 1, 510, 1, 6, 1, 1919190, 1, 6, 1, 13530, 1, 1806, 1, 690, 1, 282, 1, 46410, 1, 66, 1, 1590, 1, 798, 1, 870, 1, 354, 1, 56786730, 1 (list)
OFFSET

0,2

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.

.....................................................

page 1

Two-dimensional Arrays of Numbers

  • Many familiar numbers are not part of of a sequence but rather belong to a two-dimensional array. These are entered into the database by reading them by rows (or antidiagonals in the case of square arrays) to convert them to one-dimensional sequences.
  • Perhaps the most familiar array is Pascal's Triangle of binomial coefficients C(n,k) = n! / (k! (n-k)!) :
1
11
121
1331
14641
15101051
1615201561
172135352171
........................

  • Reading this by rows produces the following sequence.

A007318 Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0<=k<=n.
(Formerly M0082)
1175
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1 (list; table; graph; refs; listen; history; edit; internal format)
OFFSET

0,5

COMMENTS

C(n,k) = number of k-element subsets of an n-element set.

Row n gives coefficients in expansion of (1+x)^n.

C(n+k-1,n-1) is the number of ways of placing k indistinguishable balls into n boxes (the "bars and stars" argument - see Feller).

C(n-1,m-1) is the number of compositions of n with m summands.

C(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0) and (1,1). [Joerg Arndt, Jul 01 2011]

If thought of as an infinite lower triangular matrix, inverse begins:

+1

-1 +1

+1 -2 +1

-1 +3 -3 +1

+1 -4 +6 -4 +1

The string of 2^n palindromic binomial coefficients starting after the A006516(n)-th entry are all odd. - Lekraj Beedassy (blekraj(AT)yahoo.com), May 20 2003

C(n+k-1,n-1) is the number of standard tableaux of shape (n,1^k). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004

Can be viewed as an array, read by antidiagonals, where the entries in the first row and column are all 1's and A(i,j) = A(i-1,j) + A(i,j-1) for all other entries. The determinants of all its n X n subarrays starting at (0,0) are all 1. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 17 2004

Also the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals j+1 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006

C(n-3,k-1) counts the permutations in S_n which have zero occurrences of the pattern 231 and one occurrence of the pattern 132 and k descents. C(n-3,k-1) also counts the permutations in S_n which have zero occurrences of the pattern 231 and one occurrence of the pattern 213 and k descents. - David Hoek (david.hok(AT)telia.com), Feb 28 2007

Inverse of A130595 (as an infinite lower triangular matrix). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 21 2007

Consider integer lists LL of lists L of the form LL=[m#L]=[m#[k#2]] (where '#' means 'times') like LL(m=3,k=3) = [[2,2,2],[2,2,2],[2,2,2]]. The number of the integer list partitions of LL(m,k) is equal to C(m+k,k) if multiple partitions like [[1,1],[2],[2]] and [[2],[2],[1,1]] and [[2],[1,1],[2]] are count only once. For the example we find 4*5*6/3! = 20 = C(6,3). - Thomas Wieder (thomas.wieder(AT)t-online.de), Oct 03 2007

The infinitesimal generator for the Pascal triangle and its inverse is A132440. - Tom Copeland (tcjpn(AT)msn.com), Nov 15 2007

Row n>=2 gives the number of k-digit (k>0) base n numbers with strictly decreasing digits; e.g. row 10 for A009995. Similarly, row n-1>=2 gives the number of k-digit (k>1) base n numbers with strictly increasing digits; see A009993 and compare A118629. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Nov 25 2007

Comments from Lee Naish (lee(AT)cs.mu.oz.au), Mar 07 2008: (Start) C(n+k-1, k) is the number of ways a sequence of length k can be partitioned into n subsequences (see the Naish link).

C(n+k-1, k) is also the number of n- (or fewer) digit numbers written in radix at least k whose digits sum to k. For example, in decimal, there are C(3+3-1,3)=10 3-digit numbers whose digits sum to 3 (see A052217) and also C(4+2-1,2)=10 4-digit numbers whose digits sum to 2 (see A052216). This relationship can be used to generate the numbers of sequences A052216 to A052224 (and further sequences using radix greater than 10). (End)

Denote by sigma_k(x_1,x_2,...,x_n) the elementary symmetric polynomials. Then: C(2n+1,2k+1)=sigma_{n-k}(x_1,x_2,...,x_n), where x_i=tan^2(i*Pi/(2n+1)),(i=1,2,...,n). C(2n,2k+1)=2n*sigma_{n-1-k}(x_1,x_2,...,x_{n-1}), where x_i=tan^2(i*Pi/(2n)), (i=1,2,...,n-1). C(2n,2k)=sigma_{n-k}(x_1,x_2,...,x_n), where x_i=tan^2((2i-1)Pi/(4n)), (i=1,2,...,n). C(2n+1,2k)=(2n+1)sigma_{n-k}(x_1,x_2,...,x_n), where x_i=tan^2((2i-1)Pi/(4n+2)), (i=1,2,...,n). - Milan R. Janjic (agnus(AT)blic.net), May 07 2008

Given matrices R and S with R(n,k) = C(n,k)*r(n-k) and S(n,k) = C(n,k)*s(n-k), then R*S = T where T(n,k) = C(n,k)*[r(.)+s(.)]^(n-k), umbrally. And, the e.g.f.s for the row polynomials of R, S and T are, respectively, exp(x*t)*exp[r(.)*x], exp(x*t)*exp[s(.)*x] and exp(x*t)*exp[r(.)*x]*exp[s(.)*x] = exp{[t+r(.)+s(.)]*x}. The row polynomials are essentially Appell polynomials. See A132382 for an example. [From Tom Copeland (tcjpn(AT)msn.com), Aug 21 2008]

Contribution from Clark Kimberling (ck6(AT)evansville.edu), Sep 15 2008: (Start)

As the rectangle R(m,n)=C(m+n-2,m-1), the weight array W (defined

generally at A114112) of R is essentially R itself, in the sense that

if row 1 and column 1 of W=A144225 are deleted, the remaining array is R. (End)

If A007318 = M as infinite lower triangular matrix, M^n gives A130595, A023531, A007318, A038207, A027465, A038231, A038243, A038255, A027466, A038279, A038291, A038303, A038315, A038327, A133371, A147716, A027467 for n=-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 11 2008]

The coefficients of the polynomials with e.g.f. exp(x*t)*(cosh(t)+sinh(t)). [From Peter Luschny , Jul 09 2009]

Contribution from Johannes W. Meijer, Sep 22 2010: (Start)

The triangle or chess sums, see A180662 for their definitions, link Pascal's triangle with twenty different sequences, see the crossrefs. All sums come in pairs due to the symmetrical nature of this triangle. The knight sums Kn14 - Kn110 have been added. It is remarkable that all knight sums are related to the Fibonacci numbers, i.e. A000045, but none of the others.

(End)

C(n,k) is also the number of ways to distribute n+1 balls into k+1 urns so that each urn gets at least one ball. See example in the example section below.

C(n,k) is the number of increasing functions from {1,...,k} to {1,...,n} since there are C(n,k) ways to choose the k distinct, ordered elements of the range from the codomain {1,...,n}. See example in the example section below. [From Dennis Walsh, April 7 2011]

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, Arxiv preprint arXiv:1105.3043, 2011

Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, Arxiv preprint arXiv:1105.3044, 2011

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 63ff.

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 4.

Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 306.

P. Curtz, Integration numerique des systemes differentiels..,C.C.S.A.,Arcueil,1969. [From Paul Curtz (bpcrtz(AT)free.fr), Mar 06 2009]

W. Feller, An Introduction to Probability Theory and Its Application, Vol. 1, 2nd ed. New York: Wiley, p. 36, 1968.

D. Fowler, The binomial coefficient function, Amer. Math. Monthly, 103 (1996), 1-17.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 155.

D. Hoek, Parvisa moenster i permutationer [Swedish], 2007.

D. E. Knuth, The Art of Computer Programming, Vol. 1, 2nd ed., p. 52.

S. K. Lando, Lecture on Generating Functions, Amer. Math. Soc., Providence, R.I., 2003, pp. 60-61.

D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study of general and palindromic compositions, Integers 4 (2004), A23, 26 pp.

Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 6.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 2.

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996, p. 143.

L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 115-8, Penguin Books 1987.

LINKS

N. J. A. Sloane, First 141 rows of Pascal's triangle, formatted as a simple linear sequence n, a(n)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

V. Asundi, Generate a Yanghui Triangle [Broken link]

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps

D. Butler, Pascal's Triangle

L. Euler, On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc.)^n

L. Euler, De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc.)^n E709

A. Farina, et al.,Tartaglia-Pascal's triangle: a historical perspective with applications(subscription required).

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

Nick Hobson, Python program

Matthew Hubbard and Tom Roby, Pascal's Triangle From Top to Bottom

S. Kak, The Golden Mean and the Physics of Aesthetics

W. Knight, Short Table of Binomial Coefficients

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Mathforum, Pascal's Triangle

Mathforum, Links for Pascal's triangle

C. McDermottroe, n-th row generator of Pascal's triangle

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees

Lee Naish Pascal's Triangle and debugging software

A. Necer, Series formelles et produit de Hadamard

G. Sivek et al., ThinkQuest, Pascal's Triangle Row Generator

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

H. Verrill, Pascal's Triangle and related triangles

G. Villemin's Almanach of Numbers, Triangle de Pascal

Eric Weisstein's World of Mathematics, Pascals Triangle

Wikipedia, Pascal's triangle

H. S. Wilf, Generatingfunctionology</a

>, 2nd edn., Academic Press, NY, 1994, pp. 12ff.

K. Williams, Mathforum, Interactive Pascal's Triangle

K. Williams, MathForum, Pascal's Triangle to Row 19

D. Zeilberger, The Combinatorial Astrology of Rabbi Abraham Ibn Ezra [math/9809136]

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

a(n, k) = C(n,k) = binomial(n, k).

C(n, k) = C(n-1, k) + C(n-1, k-1).

a(n+1, m) = a(n, m)+a(n, m-1), a(n, -1) := 0, a(n, m) := 0, n<m; a(0, 0)=1.

C(n, k)=n!/(k!(n-k)!) if 0<=k<=n, otherwise 0.

G.f.: 1/(1-y-x*y)=Sum(C(n, k)*x^k*y^n, n, k>=0)

G.f.: 1/(1-x-y)=Sum(C(n+k, k)*x^k*y^n, n, k>=0).

G.f. for row n: (1+x)^n = sum(k=0..n, C(n, k)x^k).

G.f. for column n: x^n/(1-x)^n.

E.g.f.: A(x, y)=exp(x+x*y).

E.g.f. for column n: x^n*exp(x)/n!.

In general the m-th power of A007318 is given by: T(0, 0) = 1, T(n, k) = T(n-1, k-1) + m*T(n-1, k), where n is the row-index and k is the column; also T(n, k) = m^(n-k) C(n, k).

Triangle T(n, k) read by rows; given by A000007 DELTA A000007, where DELTA is Deleham's operator defined in A084938.

With P(n+1) = the number of integer partitions of (n+1), p(i) = the number of parts of the i-th partition of (n+1), d(i) = the number of different parts of the i-th partition of (n+1), m(i, j) = multiplicity of the j-th part of the i-th partition of (n+1), sum_[p(i)=k]_{i=1}^{P(n+1)} = sum running from i=1 to i=P(n+1) but taking only partitions with p(i)=(k+1) parts into account, prod_{j=1}^{d(i)} = product running from j=1 to j=d(i) one has B(n, k) = sum_[p(i)=(k+1)]_{i=1}^{P(n+1)} 1/prod_{j=1}^{d(i)} m(i, j)! E.g. B(5, 3) = 10 because n=6 has the following partitions with m=3 parts: (114), (123), (222). For their multiplicities one has: (114): 3!/(2!*1!) = 3, (123): 3!/(1!*1!*1!) = 6, (222): 3!/3! = 1. The sum is 3+6+1=10=B(5, 3). - Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 03 2005

C(n, k) = Sum_{j, 0<=j<=k} = (-1)^j*C(n+1+j, k-j)*A000108(j) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 10 2005

G.f.: 1 + x(1 + x) + x^3(1 + x)^2 + x^6(1 + x)^3 + ... . - Michael Somos Sep 16 2006

Sum_{k, 0<=k<=[n/2]} x^(n-k)*T(n-k,k)= A000007(n), A000045(n+1), A002605(n), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively . Sum_{k, 0<=k<=[n/2]} (-1)^k*x^(n-k)*T(n-k,k)= A000007(n), A010892(n), A009545(n+1), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n), A084329(n+1) for x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 16 2006

C(n,k) <= A062758(n) for n > 1. - Reinhard Zumkeller, Mar 04 2008

C(t+p-1, t) = Sum(i=0..t, C(i+p-2, i)) = Sum(i=1..p, C(i+t-2, t-1)) A binomial number is the sum of its left parent and all its right ancestors, which equals the sum of its right parent and all its left ancestors. - Lee Naish (lee(AT)cs.mu.oz.au), Mar 07 2008

Contribution from Paul D. Hanna, Mar 24 2011:  (Start)

Let A(x) = Sum_{n>=0} x^(n(n+1)/2)*(1+x)^n be the g.f. of the flattened triangle:

A(x) = 1 + (x + x^2) + (x^3 + 2*x^4 + x^5) + (x^6 + 3*x^7 + 3*x^8 + x^9) +...

then A(x) equals the series Sum_{n>=0} (1+x)^n*x^n*Product_{k=1..n} (1-(1+x)*x^(2k-1))/(1-(1+x)*x^(2k));

also, A(x) equals the continued fraction 1/(1- x*(1+x)/(1+ x*(1-x)*(1+x)/(1- x^3*(1+x)/(1+ x^2*(1-x^2)*(1+x)/(1- x^5*(1+x)/(1+ x^3*(1-x^3)*(1+x)/(1- x^7*(1+x)/(1+ x^4*(1-x^4)*(1+x)/(1- ...))))))))).

These formulae are due to (1) a q-series identity and (2) a partial elliptic theta function expression. (End)

EXAMPLE

Triangle begins:

1

1, 1

1, 2, 1

1, 3, 3, 1

1, 4, 6, 4, 1

1, 5, 10, 10, 5, 1

1, 6, 15, 20, 15, 6, 1

1, 7, 21, 35, 35, 21, 7, 1

1, 8, 28, 56, 70, 56, 28, 8, 1

1, 9, 36, 84, 126, 126, 84, 36, 9, 1

1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

For example, there are C(4,2)=6 ways to distribute 5 balls BBBBB, among 3 different urns, < > ( ) [ ], so that each urn gets at least one ball, namely, <BBB>(B)[B], <B>(BBB)[B], <B>(B)[BBB], <BB>(BB)[B], <BB>(B)[BB], and <B>(BB)[BB].

For example, there are C(4,2)=6 increasing functions from {1,2} to {1,2,3,4}, namely, {(1,1),(2,2)},{(1,1),(2,3)}, {(1,1),(2,4)}, {(1,2),(2,3)}, {(1,2),(2,4)}, and {(1,3),(2,4)}. [From Dennis Walsh, April 7 2011]

MAPLE

A007318 := (n, k)->binomial(n, k);

with(combstruct):for n from 0 to 11 do seq(count(Combination(n), size=m), m = 0 .. n) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2007

MATHEMATICA

Flatten[Table[Binomial[n, k], {n, 0, 11}, {k, 0, n}]] (from Robert G. Wilson v, Jan 19 2004)

PROG

(AXIOM) -- (start)

)set expose add constructor OutputForm

pascal(0, n) == 1

pascal(n, n) == 1

pascal(i, j | 0 < i and i < j) == pascal(i-1, j-1) + pascal(i, j-1)

pascalRow(n) == [pascal(i, n) for i in 0..n]

displayRow(n) == output center blankSeparate pascalRow(n)

for i in 0..20 repeat displayRow i -- (end)

(PARI) C(n, k)=if(k<0|k>n, 0, n!/k!/(n-k)!)

(PARI) C(n, k)=if(n<0, 0, polcoeff((1+x)^n, k))

(PARI) C(n, k)=if(k<0|k>n, 0, if(k==0&n==0, 1, C(n-1, k)+C(n-1, k-1)))

(PARI) C(n, k)=binomial(n, k) \\ Charles R Greathouse IV, Jun 08 2011

(Python) See Hobson link.

Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 22 2010: (Start)

(Other) Haskell

-- Cf. http://www.haskell.org/haskellwiki/Blow_your_mind#Mathematical_sequences

pascal = iterate (\rs -> zipWith (+) ([0] ++ rs) (rs ++ [0])) [1]

a007318_list = concat pascal

a007318 n k = pascal !! n !! k

-- eop. (End)

(Maxima) create_list(binomial(n, k), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */

(PARI) /* From a g.f. of the flattened triangle (Paul Hanna): */

{a(n)=polcoeff(sum(m=0, n, (x+x^2)^m*prod(k=1, m, (1-(1+x)*x^(2*k-1))/(1-(1+x+x*O(x^n))*x^(2*k)))), n)}

CROSSREFS

Equals differences between consecutive terms of A102363 - David G. Williams (davidwilliams(AT)Paxway.com), Jan 23 2006

Cf. A047999, A026729, A052553. Row sums give A000079 (powers of 2).

Cf. A083093 (triangle read mod 3).

Partial sums of rows give triangle A008949.

Infinite matrix squared: A038207, cubed: A027465

Cf. A101164. If rows are sorted we get A061554 or A107430.

Another version: A108044.

Cf. A008277.

Cf. A132311, A132312.

Cf. A052216, A052217, A052218, A052219, A052220, A052221, A052222, A052223.

Cf. A144225. [From Clark Kimberling, Sep 15 2008]

Contribution from Johannes W. Meijer, Sep 22 2010: (Start)

Triangle sums (see the comments): A000079 (Row1); A000007 (Row2); A000045 (Kn11 & Kn21); A000071 (Kn12 & Kn22); A001924 (Kn13 & Kn23); A014162 (Kn14 & Kn24); A014166 (Kn15 & Kn25); A053739 (Kn16 & Kn26); A053295 (Kn17 & Kn27); A053296 (Kn18 & Kn28); A053308 (Kn19 & Kn29); A053309 (Kn110 & Kn210); A001519 (Kn3 & Kn4); A011782 (Fi1 & Fi2); A000930 (Ca1 & Ca2); A052544 (Ca3 & Ca4); A003269 (Gi1 & Gi2); A055988 (Gi3 & Gi4); A034943 (Ze1 & Ze2); A005251 (Ze3 & Ze4).

(End)

Sequence in context: A118433 * A108086 A130595 A108363 A076831 A119724

Adjacent sequences:  A007315 A007316 A007317 * A007319 A007320 A007321

KEYWORD

nonn,tabl,nice,easy,core

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)

  • The keyword "tabl" indicates a sequence formed from a triangular or square array of numbers.
    Clicking on this keyword produces a two-dimensional display of the sequence:

A007318 formatted as a triangular array:

1
11
121
1331
14641
15101051
1615201561
172135352171
18285670562881
193684126126843691
1104512021025221012045101
1115516533046246233016555111

Important Constants

  • The database also contains many important constants (e, pi, the speed of light, etc.).
  • These are usually represented in two ways:
    • as a sequence of decimal digits, and
    • as a continued fraction expansion
    The former is the more familiar representation, while the latter is less dependent on our base 10 decimal notation.
  • For example, here are the two sequences representing the classical constant

    Sum

First the decimal expansion:

A013661 Decimal expansion of zeta(2) = Pi^2/6. 34
1, 6, 4, 4, 9, 3, 4, 0, 6, 6, 8, 4, 8, 2, 2, 6, 4, 3, 6, 4, 7, 2, 4, 1, 5, 1, 6, 6, 6, 4, 6, 0, 2, 5, 1, 8, 9, 2, 1, 8, 9, 4, 9, 9, 0, 1, 2, 0, 6, 7, 9, 8, 4, 3, 7, 7, 3, 5, 5, 5, 8, 2, 2, 9, 3, 7, 0, 0, 0, 7, 4, 7, 0, 4, 0, 3, 2, 0, 0, 8, 7, 3, 8, 3, 3, 6, 2, 8, 9, 0, 0, 6, 1, 9, 7, 5, 8, 7, 0 (list; constant; graph; refs; listen; history; edit; internal format)
OFFSET

1,2

COMMENTS

Sum_{m = 1..inf } 1/m^2.

"In 1736 he [Leonard Euler, 1707-1783] discovered the limit to the infinite series, Sum 1/n^2. He did it by doing some rather ingenious mathematics using trigonometric functions that proved the series summed to exactly Pi^2/6. How can this be? ... This demonstrates one of the most startling characteristics of mathematics - the interconnectedness of, seemingly, unrelated ideas." - Clawson

Also dilogarithm(1). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 21 2004

Also Integral_{x=0..inf} x/(exp(x)-1).

For the partial sums see the fractional sequence A007406/A007407.

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

R. Calinger, "Leonard Euler: The First St. Petersburg Years (1727-1741)," Historia Mathematica, Vol. 23, 1996, pp. 121-166.

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 97.

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington, D.C., 1999, p. xxii.

G. F. Simmons, Calculus Gems, Section B.15,B.24 pp. 270-1,323-5 McGraw Hill 1992.

A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 261.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.

LINKS

Harry J. Smith, Table of n, a(n) for n=1,...,20000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

D. H. Bailey, J. M. Borwein and D. M. Bradley, Experimental determination of Ap'ery-like identities for zeta(4n+2)

R. Chapman, Evaluating Zeta(2):14 Proofs to Zeta(2)= (pi)^2/6

R. W. Clickery, Probability of two numbers being coprime [Broken link]

L. Euler, On the sums of series of reciprocals

L. Euler, De summis serierum reciprocarum, E41.

Math. Reference Project, The Zeta Function, Zeta(2)

Math. Reference Project, The Zeta Function, Odds That Two Numbers Are Coprime"

J. Perry, Prime Product Paradox

S. Plouffe, Plouffe's Inverter, Zeta(2) or Pi**2/6 to 100000 digits

S. Plouffe, Zeta(2) or Pi**2/6 to 10000 places

A. L. Robledo, PlanetMath.org, value of the Riemann zeta function at s=2

E. Sandifer, How Euler Did It, Estimating the Basel Problem

E. Sandifer, How Euler Did It, Basel Problem with Integrals

C. Tooth, Pi squared over six

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

Eric Weisstein's World of Mathematics, Dilogarithm MathWorld page

H. Wilf, Accelerated series for universal constants, by the WZ method

FORMULA

Limit(n-->+oo) of (1/n)*(sum(k=1, n, frac((n/k)^(1/2)))) = zeta(2) and in general have limit(n-->+oo) of (1/n)*(sum(k=1, n, frac((n/k)^(1/m)))) = zeta(m), m >= 2. - Yalcin Aktar (aktaryalcin(AT)msn.com), Jul 14 2005

EXAMPLE

1.6449340668482264364724151666460251892189499012067984377355582293700074704032...

MATHEMATICA

RealDigits[N[Pi^2/6, 100]][[1]]

PROG

(PARI) \p 200; Pi^2/6

(PARI) \p 200 dilog(1) \p 200 zeta(2)

(PARI) a(n)=if(n<1, 0, default(realprecision, n+2); floor(Pi^2/6*10^(n-1))%10)

(PARI) { default(realprecision, 20080); x=Pi^2/6; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b013661.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 29 2009]

CROSSREFS

Cf. A013679, A013631, A013680, 1/A059956.

Sequence in context: A195359 A021612 A110756 * A019174 A019166 A058158

Adjacent sequences:  A013658 A013659 A013660 * A013662 A013663 A013664

KEYWORD

cons,nonn,nice,changed

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com).

EXTENSIONS

Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009

Then the continued fraction expansion:

A013679 Continued fraction for zeta(2) = Pi^2/6. 4
1, 1, 1, 1, 4, 2, 4, 7, 1, 4, 2, 3, 4, 10, 1, 2, 1, 1, 1, 15, 1, 3, 6, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 3, 1, 1, 5, 1, 2, 2, 1, 1, 6, 27, 20, 3, 97, 105, 1, 1, 1, 1, 1, 45, 2, 8, 19, 1, 4, 1, 1, 3, 1, 2, 1, 1, 1, 5, 1, 1, 2, 3, 6, 1, 1, 1, 2, 1, 5, 1, 1, 2, 9, 5, 3, 2, 1, 1, 1 (list; graph; refs; listen; history; edit; internal format)
OFFSET

1,5

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

G. Xiao, Contfrac

Index entries for continued fractions for constants

EXAMPLE

1.6449340668482264364724151666460251892189...

1.644934066848226436472415166... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...)))) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 29 2009]

MATHEMATICA

ContinuedFraction[ Pi^2/6, 100]

PROG

(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^2/6); for (n=1, 20000, write("b013679.txt", n, " ", x[n])); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 29 2009]

CROSSREFS

Cf. A013661.

Sequence in context: A007005 A066978 A114566 * A096428 A091007 A180156

Adjacent sequences:  A013676 A013677 A013678 * A013680 A013681 A013682

KEYWORD

nonn,cofr,nice,easy

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com).

  • Many other examples of important constants may be found by consulting the Index.

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Last modified April 25 19:05 EDT 2017. Contains 285426 sequences.