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A003506 Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1<=k<=n. 57
1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 30, 20, 5, 6, 30, 60, 60, 30, 6, 7, 42, 105, 140, 105, 42, 7, 8, 56, 168, 280, 280, 168, 56, 8, 9, 72, 252, 504, 630, 504, 252, 72, 9, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 11, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Array 1/Beta(n,m) read by antidiagonals. - Michael Somos, Feb 05 2004

a(n,3) = A027480(n-2); a(n,4) = A033488(n-3). - Ross La Haye, Feb 13 2004

a(n,k) = total size of all of the elements of the family of k-size subsets of an n-element set. For example, a 2-element set, say, {1,2}, has 3 families of k-size subsets: one with 1 0-size element, one with 2 1-size elements and one with 1 2-size element; respectively, {{}}, {{1},{2}}, {{1,2}}. - Ross La Haye, Dec 31 2006

Second slice along the 1-2-plane in the cube a(m,n,o) = a(m-1,n,o) + a(m,n-1,o) + a(m,n,o-1) with a(1,0,0)=1 and a(m<>1=0,n>=0,0>=o)=0, for which the first slice is Pascal's triangle (slice read by antidiagonals). - Thomas Wieder, Aug 06 2006

Triangle, read by rows, given by [2,-1/2,1/2,0,0,0,0,0,0,...] DELTA [2,-1/2,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007

This sequence * [1/1, 1/2, 1/3, ...] = (1, 3, 7, 15, 31, ...). - Gary W. Adamson, Nov 14 2007

n-th row = coefficients of first derivative of corresponding Pascal's triangle row. Example: x^4 + 4x^3 + 6x^2 + 4x + 1 becomes (4, 12, 12, 4). - Gary W. Adamson, Dec 27 2007

T(n,5)=T(n,n-4)=A174002(n-4) for n>4; T(2*n,n)=T(2*n,n+1)=A005430(n). - Reinhard Zumkeller, Mar 05 2010

From Paul Curtz, Jun 03 2011: (Start)

Consider

   1     1/2   1/3    1/4    1/5

  -1/2  -1/6  -1/12  -1/20  -1/30

   1/3   1/12  1/30   1/60   1/105

  -1/4  -1/20 -1/60  -1/140 -1/280

   1/5   1/30  1/105  1/280  1/630

This is an autosequence (the inverse binomial transform is the sequence signed) of second kind: the main diagonal is the double of the first upper diagonal.

Note that 2, 12, 60, ... = A005430(n+1), Apery numbers = 2*A002457(n). (End)

From Louis Conover (for the 9th grade G1c mathematics class at the Chengdu Confucius International School), Mar 02 2015: (Start)

The i-th order differences of n^-1 appear in the (i+1)th row.

1,    1/2,   1/3,   1/4,    1/5,    1/6,    1/7,     1/8, ...

1/2,  1/6,  1/12,  1/20,   1/30,   1/42,   1/56,    1/72, ...

1/3, 1/12,  1/30,  1/60,  1/105,  1/168,  1/252,   1/360, ...

1/4, 1/20,  1/60, 1/140,  1/280,  1/504,  1/840,  1/1320, ...

1/5, 1/30, 1/105, 1/280,  1/630, 1/1260, 1/2310,  1/3960, ...

1/6, 1/42, 1/168, 1/504, 1/1260, 1/2772, 1/5544, 1/12012, ...

(End)

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, see 130.

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. 38.

G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

M. Elkadi and B. Mourrain, Symbolic-numeric methods for solving polynomial equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris, eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168. See p. 152.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 35.

LINKS

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

H. J. Brothers, Pascal's Prism: Supplementary Material.

D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle

FORMULA

a(n, 1) = 1/n; a(n, k) = a(n-1, k-1)-a(n, k-1) for k>1.

Considering the integer values (rather than unit fractions): a(n, k) = k*C(n, k) = n*C(n-1, k-1) = a(n, k-1)*a(n-1, k-1)/(a(n, k-1)-a(n-1, k-1)) = a(n-1, k) + a(n-1, k-1)*k/(k-1) = (a(n-1, k) + a(n-1, k-1))*n/(n-1) = k*A007318(n, k) = n*A007318(n-1, k-1). Row sums of integers are n*2^(n-1) = A001787(n); row sums of the unit fractions are A003149(n-1)/A000142(n). - Henry Bottomley, Jul 22 2002

G.f.: x*y/(1-x-y*x)^2. E.g.f: x*y*exp(x+x*y). - Vladeta Jovovic, Nov 01 2003

T(n,k) = n*binomial(n-1,k-1)=  n*A007318(n-1,k-1). - Philippe Deléham, Aug 04 2006

Binomial transform of A128064(unsigned). - Gary W. Adamson, Aug 29 2007

t(n,m) = Gamma[n]/(Gamma[n - m]*Gamma[m]. - Roger L. Bagula and Gary W. Adamson, Sep 14 2008

f[s,n] = Integrate[Exp[ -s*x]*x^n,{x,0,Infinity}]=Gamma[n]/s^n; t(n,m)=f[s,n]/(f[s,n-m]*f[s,m])=Gamma[n]/(Gamma[n - m]*Gamma[m]; the powers of s cancel out. - Roger L. Bagula and Gary W. Adamson, Sep 14 2008

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) - T(n-2,k-2), T(1,1) = 1 and, for n>1, T(n,k) = 0 if k<=1 or if k>n. - Philippe Deléham, Mar 17 2012

T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+1-k,k+1-i). - Mircea Merca, Apr 11 2012

If we include a main diagonal of zeros so that the array is in the form

  0

  1   0

  2   2   0

  3   6   3   0

  4  12  12   4   0

...

then we obtain the exponential Riordan array [x*exp(x),x], which factors as [x,x]*[exp(x),x] = A132440*A007318. This array is the infinitesimal generator for A116071. A signed version of the array is the infinitesimal generator for A215652. - Peter Bala, Sep 14 2012

a(n,k) = (n-1)!/((n-k)!(k-1)!) if k>n/2 and a(n,k)=(n-1)!/((n-k-1)!k!) otherwise. [Forms 'core' for Pascal's recurrence; gives common term of RHS of T(n,k) = T(n-1,k-1) + T(n-1,k)]. - Jon Perry, Oct 08 2013

EXAMPLE

1/1; 1/2, 1/2; 1/3, 1/6, 1/3; 1/4, 1/12, 1/12, 1/4; 1/5, 1/20, 1/30, 1/20, 1/5; ...

Triangle begins:

   1

   2   2

   3   6   3

   4  12  12    4

   5  20  30   20    5

   6  30  60   60   30    6

   7  42 105  140  105   42    7

   8  56 168  280  280  168   56    8

   9  72 252  504  630  504  252   72   9

  10  90 360  840 1260 1260  840  360  90  10

  11 110 495 1320 2310 2772 2310 1320 495 110 11

MAPLE

with(combstruct):for n from 0 to 11 do seq(m*count(Combination(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Apr 09 2008

A003506 := (n, k) -> k*binomial(n, k):

seq(print(seq(A003506(n, k), k=1..n)), n=1..7); # Peter Luschny, May 27 2011

MATHEMATICA

L[n_, 1] := 1/n; L[n_, m_] := L[n, m] = L[n - 1, m - 1] - L[n, m - 1]; Take[ Flatten[ Table[ 1 / L[n, m], {n, 1, 12}, {m, 1, n}]], 66]

t[n_, m_] = Gamma[n]/(Gamma[n - m]*Gamma[m]); Table[Table[t[n, m], {m, 1, n - 1}], {n, 2, 12}]; Flatten[%] (* Roger L. Bagula and Gary W. Adamson, Sep 14 2008 *)

Table[k*Binomial[n, k], {n, 1, 7}, {k, 1, n}] (* Peter Luschny, May 27 2011 *)

t[n_, k_] := Denominator[n!*k!/(n+k+1)!]; Table[t[n-k, k] , {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)

PROG

(PARI) A(i, j)=if(i<1||j<1, 0, 1/subst(intformal(x^(i-1)*(1-x)^(j-1)), x, 1))

(PARI) A(i, j)=if(i<1||j<1, 0, 1/sum(k=0, i-1, (-1)^k*binomial(i-1, k)/(j+k)))

(PARI) {T(n, k) = (n + 1 - k) * binomial( n, k - 1)} /* Michael Somos, Feb 06 2011 */

(Haskell)

a003506 n k = a003506_tabl !! (n-1) !! (n-1)

a003506_row n = a003506_tabl !! (n-1)

a003506_tabl = scanl1 (\xs ys ->

   zipWith (+) (zipWith (+) ([0] ++ xs) (xs ++ [0])) ys) a007318_tabl

a003506_list = concat a003506_tabl

-- Reinhard Zumkeller, Nov 14 2013, Nov 17 2011

(Sage)

T_row = lambda n: (n*(x+1)^(n-1)).list()

for n in (1..10): print T_row(n) # Peter Luschny, Feb 04 2017

CROSSREFS

Cf. A007622, A128064, A164555/A027642.

Cf. A094305, A121547, A121306, A119800, A002378, A007318.

Row sums are in A001787. Central column is A002457. Half-diagonal is in A090816. A116071, A215652.

Denominators of i-th order differences of n^-1 are given in: (1st) A002378, (2nd) A027480, (3rd) A033488, (4th) A174002, (5th) A253946. - Louis Conover, Mar 02 2015

Sequence in context: A296320 A296396 A125102 * A047662 A329655 A183474

Adjacent sequences:  A003503 A003504 A003505 * A003507 A003508 A003509

KEYWORD

tabl,nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by N. J. A. Sloane, Oct 07 2007

STATUS

approved

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Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)