login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A028246 Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows. 63
1, 1, 1, 1, 3, 2, 1, 7, 12, 6, 1, 15, 50, 60, 24, 1, 31, 180, 390, 360, 120, 1, 63, 602, 2100, 3360, 2520, 720, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320, 1, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 362880 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Let M = n X n matrix with (i,j)-th entry a(n+1-j, n+1-i), e.g., if n = 3, M = [1 1 1; 3 1 0; 2 0 0]. Given a sequence s = [s(0)..s(n-1)], let b = [b(0)..b(n-1)] be its inverse binomial transform and let c = [c(0)..c(n-1)] = M^(-1)*transpose(b). Then s(k) = Sum_{i=0..n-1} b(i)*binomial(k,i) = Sum_{i=0..n-1} c(i)*k^i, k=0..n-1. - Gary W. Adamson, Nov 11 2001
From Gary W. Adamson, Aug 09 2008: (Start)
Julius Worpitzky's 1883 algorithm generates Bernoulli numbers.
By way of example [Wikipedia]:
B0 = 1;
B1 = 1/1 - 1/2;
B2 = 1/1 - 3/2 + 2/3;
B3 = 1/1 - 7/2 + 12/3 - 6/4;
B4 = 1/1 - 15/2 + 50/3 - 60/4 + 24/5;
B5 = 1/1 - 31/2 + 180/3 - 390/4 + 360/5 - 120/6;
B6 = 1/1 - 63/2 + 602/3 - 2100/4 + 3360/5 - 2520/6 + 720/7;
...
Note that in this algorithm, odd n's for the Bernoulli numbers sum to 0, not 1, and the sum for B1 = 1/2 = (1/1 - 1/2). B3 = 0 = (1 - 7/2 + 13/3 - 6/4) = 0. The summation for B4 = -1/30. (End)
Pursuant to Worpitzky's algorithm and given M = A028246 as an infinite lower triangular matrix, M * [1/1, -1/2, 1/3, ...] (i.e., the Harmonic series with alternate signs) = the Bernoulli numbers starting [1/1, 1/2, 1/6, ...]. - Gary W. Adamson, Mar 22 2012
From Tom Copeland, Oct 23 2008: (Start)
G(x,t) = 1/(1 + (1-exp(x*t))/t) = 1 + 1 x + (2 + t)*x^2/2! + (6 + 6t + t^2)*x^3/3! + ... gives row polynomials for A090582, the f-polynomials for the permutohedra (see A019538).
G(x,t-1) = 1 + 1*x + (1 + t)*x^2 / 2! + (1 + 4t + t^2)*x^3 / 3! + ... gives row polynomials for A008292, the h-polynomials for permutohedra.
G[(t+1)x,-1/(t+1)] = 1 + (1+ t) x + (1 + 3t + 2 t^2) x^2 / 2! + ... gives row polynomials for the present triangle. (End)
The Worpitzky triangle seems to be an apt name for this triangle. - Johannes W. Meijer, Jun 18 2009
If Pascal's triangle is written as a lower triangular matrix and multiplied by A028246 written as an upper triangular matrix, the product is a matrix where the (i,j)-th term is (i+1)^j. For example,
1,0,0,0 1,1,1, 1 1,1, 1, 1
1,1,0,0 * 0,1,3, 7 = 1,2, 4, 8
1,2,1,0 0,0,2,12 1,3, 9,27
1,3,3,1 0,0,0, 6 1,4,16,64
So, numbering all three matrices' rows and columns starting at 0, the (i,j) term of the product is (i+1)^j. - Jack A. Cohen (ProfCohen(AT)comcast.net), Aug 03 2010
The Fi1 and Fi2 triangle sums are both given by sequence A000670. For the definition of these triangle sums see A180662. The mirror image of the Worpitzky triangle is A130850. - Johannes W. Meijer, Apr 20 2011
Let S_n(m) = 1^m + 2^m + ... + n^m. Then, for n >= 0, we have the following representation of S_n(m) as a linear combination of the binomial coefficients:
S_n(m) = Sum_{i=1..n+1} a(i+n*(n+1)/2)*C(m,i). E.g., S_2(m) = a(4)*C(m,1) + a(5)*C(m,2) + a(6)*C(m,3) = C(m,1) + 3*C(m,2) + 2*C(m,3). - Vladimir Shevelev, Dec 21 2011
Given the set X = [1..n] and 1 <= k <= n, then a(n,k) is the number of sets T of size k of subset S of X such that S is either empty or else contains 1 and another element of X and such that any two elemements of T are either comparable or disjoint. - Michael Somos, Apr 20 2013
Working with the row and column indexing starting at -1, a(n,k) gives the number of k-dimensional faces in the first barycentric subdivision of the standard n-dimensional simplex (apply Brenti and Welker, Lemma 2.1). For example, the barycentric subdivision of the 2-simplex (a triangle) has 1 empty face, 7 vertices, 12 edges and 6 triangular faces giving row 4 of this triangle as (1,7,12,6). Cf. A053440. - Peter Bala, Jul 14 2014
See A074909 and above g.f.s for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014
An e.g.f. G(x,t) = exp[P(.,t)x] = 1/t - 1/[t+(1-t)(1-e^(-xt^2))] = (1-t) * x + (-2t + 3t^2 - t^3) * x^2/2! + (6t^2 - 12t^3 + 7t^4 - t^5) * x^3/3! + ... for the shifted, reverse, signed polynomials with the first element nulled, is generated by the infinitesimal generator g(u,t)d/du = [(1-u*t)(1-(1+u)t)]d/du, i.e., exp[x * g(u,t)d/du] u eval. at u=0 generates the polynomials. See A019538 and the G. Rzadkowski link below for connections to the Bernoulli and Eulerian numbers, a Ricatti differential equation, and a soliton solution to the KdV equation. The inverse in x of this e.g.f. is Ginv(x,t) = (-1/t^2)*log{[1-t(1+x)]/[(1-t)(1-tx)]} = [1/(1-t)]x + [(2t-t^2)/(1-t)^2]x^2/2 + [(3t^2-3t^3+t^4)/(1-t)^3]x^3/3 + [(4t^3-6t^4+4t^5-t^6)/(1-t)^4]x^4/4 + ... . The numerators are signed, shifted A135278 (reversed A074909), and the rational functions are the columns of A074909. Also, dG(x,t)/dx = g(G(x,t),t) (cf. A145271). (Analytic G(x,t) added, and Ginv corrected and expanded on Dec 28 2015.) - Tom Copeland, Nov 21 2014
The operator R = x + (1 + t) + t e^{-D} / [1 + t(1-e^(-D))] = x + (1+t) + t - (t+t^2) D + (t+3t^2+2t^3) D^2/2! - ... contains an e.g.f. of the reverse row polynomials of the present triangle, i.e., A123125 * A007318 (with row and column offset 1 and 1). Umbrally, R^n 1 = q_n(x;t) = (q.(0;t)+x)^n, with q_m(0;t) = (t+1)^(m+1) - t^(m+1), the row polynomials of A074909, and D = d/dx. In other words, R generates the Appell polynomials associated with the base sequence A074909. For example, R 1 = q_1(x;t) = (q.(0;t)+x) = q_1(0;t) + q__0(0;t)x = (1+2t) + x, and R^2 1 = q_2(x;t) = (q.(0;t)+x)^2 = q_2(0:t) + 2q_1(0;t)x + q_0(0;t)x^2 = 1+3t+3t^2 + 2(1+2t)x + x^2. Evaluating the polynomials at x=0 regenerates the base sequence. With a simple sign change in R, R generates the Appell polynomials associated with A248727. - Tom Copeland, Jan 23 2015
For a natural refinement of this array, see A263634. - Tom Copeland, Nov 06 2015
From Wolfdieter Lang, Mar 13 2017: (Start)
The e.g.f. E(n, x) for {S(n, m)}_{m>=0} with S(n, m) = Sum_{k=1..m} k^n, n >= 0, (with undefined sum put to 0) is exp(x)*R(n+1, x) with the exponential row polynomials R(n, x) = Sum_{k=1..n} a(n, k)*x^k/k!. E.g., e.g.f. for n = 2, A000330: exp(x)*(1*x/1!+3*x^2/2!+2*x^3/3!).
The o.g.f. G(n, x) for {S(n, m)}_{m >=0} is then found by Laplace transform to be G(n, 1/p) = p*Sum_{k=1..n} a(n+1, k)/(p-1)^(2+k).
Hence G(n, x) = x/(1 - x)^(n+2)*Sum_{k=1..n} A008292(n,k)*x^(k-1).
E.g., n=2: G(2, 1/p) = p*(1/(p-1)^2 + 3/(p-1)^3 + 2/(p-1)^4) = p^2*(1+p)/(p-1)^4; hence G(2, x) = x*(1+x)/(1-x)^4.
This works also backwards: from the o.g.f. to the e.g.f. of {S(n, m)}_{m>=0}. (End)
a(n,k) is the number of k-tuples of pairwise disjoint and nonempty subsets of a set of size n. - Dorian Guyot, May 21 2019
From Rajesh Kumar Mohapatra, Mar 16 2020: (Start)
a(n-1,k) is the number of chains of length k in a partially ordered set formed from subsets of an n-element set ordered by inclusion such that the first term of the chains is either the empty set or an n-element set.
Also, a(n-1,k) is the number of distinct k-level rooted fuzzy subsets of an n-set ordered by set inclusion. (End)
The relations on p. 34 of Hasan (also p. 17 of Franco and Hasan) agree with the relation between A019538 and this entry given in the formula section. - Tom Copeland, May 14 2020
T(n,k) is the size of the Green's L-classes in the D-classes of rank (k-1) in the semigroup of partial transformations on an (n-1)-set. - Geoffrey Critzer, Jan 09 2023
T(n,k) is the number of strongly connected binary relations on [n] that have period k (A367948) and index 1. See Theorem 5.4.25(6) in Ki Hang Kim reference. - Geoffrey Critzer, Dec 07 2023
REFERENCES
Ki Hang Kim, Boolean Matrix Theory and Applications, Marcel Dekker, New York and Basel (1982).
LINKS
V. S. Abramovich, Power sums of natural numbers, Kvant, no. 5 (1973), 22-25. (in Russian)
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
H. Belbachir, M. Rahmani, and B. Sury, Sums Involving Moments of Reciprocals of Binomial Coefficients, J. Int. Seq. 14 (2011) #11.6.6.
Hacene Belbachir and Mourad Rahmani, Alternating Sums of the Reciprocals of Binomial Coefficients, Journal of Integer Sequences, Vol. 15 (2012), #12.2.8.
F. Brenti and V. Welker, f-vectors of barycentric subdivisions, arXiv:math/0606356v1 [math.CO], Math. Z., 259(4), 849-865, 2008.
Patibandla Chanakya and Putla Harsha, Generalized Nested Summation of Powers of Natural Numbers, arXiv:1808.08699 [math.NT], 2018. See Table 1.
Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
E. Delucchi, A. Pixton and L. Sabalka. Face vectors of subdivided simplicial complexes arXiv:1002.3201v3 [math.CO], Discrete Mathematics, Volume 312, Issue 2, January 2012, Pages 248-257.
G. H. E Duchamp, N. Hoang-Nghia, and A. Tanasa, A word Hopf algebra based on the selection/quotient principle, arXiv:1207.6522 [math.CO], 2012-2013; Séminaire Lotharingien de Combinatoire 68 (2013), Article B68c.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry , arXiv preprint arXiv:1904.07954 [hep-th], 2019
A. Hasan, Physics and Mathematics of Graded Quivers, dissertation, Graduate Center, City University of New York, 2019.
H. Hasse, Ein Summierungsverfahren für die Riemannsche Zeta-Reihe, Math. Z. 32, 458-464 (1930).
Guy Louchard, Werner Schachinger, and Mark Daniel Ward, The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis, arXiv:2203.14773 [math.PR], 2022. See p. 5.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20(1) (2013), P11.
Richard J. Mathar, Integrals Associated with the Digamma Integral Representation, arXiv:2308.14154 [math.GM], 2023. See p. 5.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
A. Riskin and D. Beckwith, Problem 10231, Amer. Math. Monthly, 102 (1995), 175-176.
G. Rzadkowski, Bernoulli numbers and solitons revisited, Journal of Nonlinear Mathematical Physics, Volume 17, Issue 1, 2010.
G. J. Simmons, A combinatorial problem associated with a family of combination locks, Math. Mag., 37 (1964), 127-132 (but there are errors). The triangle is on page 129.
N. J. A. Sloane, Transforms
Sam Vandervelde, The Worpitzky Numbers Revisited, Amer. Math. Monthly, 125:3 (2018), 198-206.
Wikipedia, Bernoulli number.
David C. Wood, The computation of polylogarithms (2014).
FORMULA
E.g.f.: -log(1-y*(exp(x)-1)). - Vladeta Jovovic, Sep 28 2003
a(n, k) = S2(n, k)*(k-1)! where S2(n, k) is a Stirling number of the second kind (cf. A008277). Also a(n,k) = T(n,k)/k, where T(n, k) = A019538.
Essentially same triangle as triangle [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is Deléham's operator defined in A084938, but the notation is different.
Sum of terms in n-th row = A000629(n) - Gary W. Adamson, May 30 2005
The row generating polynomials P(n, t) are given by P(1, t)=t, P(n+1, t) = t(t+1)(d/dt)P(n, t) for n >= 1 (see the Riskin and Beckwith reference). - Emeric Deutsch, Aug 09 2005
From Gottfried Helms, Jul 12 2006: (Start)
Delta-matrix as can be read from H. Hasse's proof of a connection between the zeta-function and Bernoulli numbers (see link below).
Let P = lower triangular matrix with entries P[row,col] = binomial(row,col).
Let J = unit matrix with alternating signs J[r,r]=(-1)^r.
Let N(m) = column matrix with N(m)(r) = (r+1)^m, N(1)--> natural numbers.
Let V = Vandermonde matrix with V[r,c] = (r+1)^c.
V is then also N(0)||N(1)||N(2)||N(3)... (indices r,c always beginning at 0).
Then Delta = P*J * V and B' = N(-1)' * Delta, where B is the column matrix of Bernoulli numbers and ' means transpose, or for the single k-th Bernoulli number B_k with the appropriate column of Delta,
B_k = N(-1)' * Delta[ *,k ] = N(-1)' * P*J * N(k).
Using a single column instead of V and assuming infinite dimension, H. Hasse showed that in x = N(-1) * P*J * N(s), where s can be any complex number and s*zeta(1-s) = x.
His theorem reads: s*zeta(1-s) = Sum_{n>=0..inf} (n+1)^-1*delta(n,s), where delta(n,s) = Sum_{j=0..n} (-1)^j * binomial(n,j) * (j+1)^s.
(End)
a(n,k) = k*a(n-1,k) + (k-1)*a(n-1,k-1) with a(n,1) = 1 and a(n,n) = (n-1)!. - Johannes W. Meijer, Jun 18 2009
Rephrasing the Meijer recurrence above: Let M be the (n+1)X(n+1) bidiagonal matrix with M(r,r) = M(r,r+1) = r, r >= 1, in the two diagonals and the rest zeros. The row a(n+1,.) of the triangle is row 1 of M^n. - Gary W. Adamson, Jun 24 2011
From Tom Copeland, Oct 11 2011: (Start)
With e.g.f.. A(x,t) = G[(t+1)x,-1/(t+1)]-1 (from 2008 comment) = -1 + 1/[1-(1+t)(1-e^(-x))] = (1+t)x + (1+3t+2t^2)x^2/2! + ..., the comp. inverse in x is
B(x,t)= -log(t/(1+t)+1/((1+t)(1+x))) = (1/(1+t))x - ((1+2t)/(1+t)^2)x^2/2 + ((1+3t+3t^2)/(1+t)^3)x^3/3 + .... The numerators are the row polynomials of A074909, and the rational functions are (omitting the initial constants) signed columns of the re-indexed Pascal triangle A007318.
Let h(x,t)= 1/(dB/dx) = (1+x)(1+t(1+x)), then the row polynomial P(n,t) = (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t), with P(1,t)=1+t. (Series added Dec 29 2015.)(End)
Let <n,k> denote the Eulerian numbers A173018(n,k), then T(n,k) = Sum_{j=0..n} <n,j>*binomial(n-j,n-k). - Peter Luschny, Jul 12 2013
Matrix product A007318 * A131689. The n-th row polynomial R(n,x) = Sum_{k >= 1} k^(n-1)*(x/(1 + x))^k, valid for x in the open interval (-1/2, inf). Cf A038719. R(n,-1/2) = (-1)^(n-1)*(2^n - 1)*Bernoulli(n)/n. - Peter Bala, Jul 14 2014
a(n,k) = A141618(n,k) / C(n,k-1). - Tom Copeland, Oct 25 2014
For the row polynomials, A028246(n,x) = A019538(n-1,x) * (1+x). - Tom Copeland, Dec 28 2015
n-th row polynomial R(n,x) = (1+x) o (1+x) o ... o (1+x) (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See example E11 in the Bala link. - Peter Bala, Jan 12 2018
From Dorian Guyot, May 21 2019: (Start)
Sum_{i=0..k} binomial(k,i) * a(n,i) = (k+1)^n.
Sum_{k=0..n} a(n,k) = 2*A000670(n).
(End)
With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of this entry, A028246, are given by x^n * A_n(1 + 1/x;0). Other specializations of A_n(x;y) give A046802, A090582, A119879, A130850, and A248727. - Tom Copeland, Jan 24 2020
The row generating polynomials R(n,x) = Sum_{i=1..n} a(n,i) * x^i satisfy the recurrence equation R(n+1,x) = R(n,x) + Sum_{k=0..n-1} binomial(n-1,k) * R(k+1,x) * R(n-k,x) for n >= 1 with initial value R(1,x) = x. - Werner Schulte, Jun 17 2021
EXAMPLE
The triangle a(n, k) starts:
n\k 1 2 3 4 5 6 7 8 9
1: 1
2: 1 1
3: 1 3 2
4: 1 7 12 6
5: 1 15 50 60 24
6: 1 31 180 390 360 120
7: 1 63 602 2100 3360 2520 720
8: 1 127 1932 10206 25200 31920 20160 5040
9: 1 255 6050 46620 166824 317520 332640 181440 40320
... [Reformatted by Wolfdieter Lang, Mar 26 2015]
-----------------------------------------------------
Row 5 of triangle is {1,15,50,60,24}, which is {1,15,25,10,1} times {0!,1!,2!,3!,4!}.
From Vladimir Shevelev, Dec 22 2011: (Start)
Also, for power sums, we have
S_0(n) = C(n,1);
S_1(n) = C(n,1) + C(n,2);
S_2(n) = C(n,1) + 3*C(n,2) + 2*C(n,3);
S_3(n) = C(n,1) + 7*C(n,2) + 12*C(n,3) + 6*C(n,4);
S_4(n) = C(n,1) + 15*C(n,2) + 50*C(n,3) + 60*C(n,4) + 24*C(n,5); etc.
(End)
For X = [1,2,3], the sets T are {{}}, {{},{1,2}}, {{},{1,3}}, {{},{1,2,3}}, {{},{1,2},{1,2,3}}, {{},{1,3},{1,2,3}} and so a(3,1)=1, a(3,2)=3, a(3,3)=2. - Michael Somos, Apr 20 2013
MAPLE
a := (n, k) -> add((-1)^(k-i)*binomial(k, i)*i^n, i=0..k)/k;
seq(print(seq(a(n, k), k=1..n)), n=1..10);
T := (n, k) -> add(eulerian1(n, j)*binomial(n-j, n-k), j=0..n):
seq(print(seq(T(n, k), k=0..n)), n=0..9); # Peter Luschny, Jul 12 2013
MATHEMATICA
a[n_, k_] = Sum[(-1)^(k-i) Binomial[k, i]*i^n, {i, 0, k}]/k; Flatten[Table[a[n, k], {n, 10}, {k, n}]] (* Jean-François Alcover, May 02 2011 *)
PROG
(PARI) {T(n, k) = if( k<0 || k>n, 0, n! * polcoeff( (x / log(1 + x + x^2 * O(x^n) ))^(n+1), n-k))}; /* Michael Somos, Oct 02 2002 */
(PARI) {T(n, k) = stirling(n, k, 2)*(k-1)!}; \\ G. C. Greubel, May 31 2019
(Sage)
def A163626_row(n) :
x = polygen(ZZ, 'x')
A = []
for m in range(0, n, 1) :
A.append((-x)^m)
for j in range(m, 0, -1):
A[j - 1] = j * (A[j - 1] - A[j])
return list(A[0])
for i in (1..7) : print(A163626_row(i)) # Peter Luschny, Jan 25 2012
(Sage) [[stirling_number2(n, k)*factorial(k-1) for k in (1..n)] for n in (1..10)] # G. C. Greubel, May 30 2019
(Magma) [[StirlingSecond(n, k)*Factorial(k-1): k in [1..n]]: n in [1..10]]; // G. C. Greubel, May 30 2019
(GAP) Flat(List([1..10], n-> List([1..n], k-> Stirling2(n, k)* Factorial(k-1) ))) # G. C. Greubel, May 30 2019
(Python) # Assuming offset (n, k) = (0, 0).
def T(n, k):
if k > n: return 0
if k == 0: return 1
return k*T(n - 1, k - 1) + (k + 1)*T(n - 1, k)
for n in range(9):
print([T(n, k) for k in range(n + 1)]) # Peter Luschny, Apr 26 2022
CROSSREFS
Dropping the column of 1's gives A053440.
Without the k in the denominator (in the definition), we get A019538. See also the Stirling number triangle A008277.
Row sums give A000629(n-1) for n >= 1.
Cf. A027642, A002445. - Gary W. Adamson, Aug 09 2008
Appears in A161739 (RSEG2 triangle), A161742 and A161743. - Johannes W. Meijer, Jun 18 2009
Binomial transform is A038719. Cf. A131689.
Cf. A119879.
From Rajesh Kumar Mohapatra, Mar 29 2020: (Start)
A000007(n-1) (column k=1), A000225(n-1) (column k=2), A028243(n-1) (column k=3), A028244(n-1) (column k=4), A028245(n-1) (column k=5), for n > 0.
Diagonal gives A000142(n-1), for n >=1.
Next-to-last diagonal gives A001710,
Third, fourth, fifth, sixth, seventh external diagonal respectively give A005460, A005461, A005462, A005463, A005464. (End)
Sequence in context: A306226 A186370 A163626 * A082038 A143774 A196842
KEYWORD
nonn,easy,nice,tabl
AUTHOR
N. J. A. Sloane, Doug McKenzie (mckfam4(AT)aol.com)
EXTENSIONS
Definition corrected by Li Guo, Dec 16 2006
Typo in link corrected by Johannes W. Meijer, Oct 17 2009
Error in title corrected by Johannes W. Meijer, Sep 24 2010
Edited by M. F. Hasler, Oct 29 2014
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 25 12:21 EDT 2024. Contains 374588 sequences. (Running on oeis4.)