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A269945

Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 21, 14, 1, 0, 1, 85, 147, 30, 1, 0, 1, 341, 1408, 627, 55, 1, 0, 1, 1365, 13013, 11440, 2002, 91, 1, 0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1

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OFFSET

0,9

COMMENTS

Also known as central factorial numbers T(2*n, 2*k) (cf. A036969).

The analog for the Stirling cycle numbers is A269944.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 1..150, flattened.)

M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.

Peter Luschny, The P-transform.

Peter Luschny, The Partition Transform - A SageMath Jupyter Notebook.

Aleks Žigon Tankosič, The Lah Numbers with Higher Level and the Lah Numbers of Order s, arXiv:2510.26842 [math.GM], 2025. See p. 4.

FORMULA

T(n, k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = 1/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

T(n, 2) = (4^(n - 1) - 1)/3 for n >= 2 (cf. A002450).

T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).

From Fabián Pereyra, Apr 25 2022: (Start)

T(n, k) = (1/(2*k)!)*Sum_{j=0..2*k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n).

T(n, k) = Sum_{j=2*k..2*n} (-k)^(2*n - j)*binomial(2*n, j)*Stirling2(j, 2*k).

T(n, k) = Sum_{j=0..2*n} (-1)^(j - k)*Stirling2(2*n - j, k)*Stirling2(j, k). (End)

T(n, k) = (2*n)! [t^(2*(n-k+1))] [x^(2*n)] (1 + t^2*(cosh(2*sinh(t*x/2)/t))). - Peter Luschny, Feb 29 2024

EXAMPLE

Triangle starts:

[0] [1]

[1] [0, 1]

[2] [0, 1, 1]

[3] [0, 1, 5, 1]

[4] [0, 1, 21, 14, 1]

[5] [0, 1, 85, 147, 30, 1]

[6] [0, 1, 341, 1408, 627, 55, 1]

MAPLE

T := proc(n, k) option remember;

`if`(n=k, 1,

`if`(k<0 or k>n, 0,

T(n-1, k-1) + k^2*T(n-1, k))) end:

for n from 0 to 9 do seq(T(n, k), k=0..n) od;

# Alternative: with the P-transform (cf. A269941):

A269945_row := n -> PTrans(n, n->`if`(n=1, 1, 1/(n*(4*n-2))), (n, k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269945_row(n)), n=0..8);

# Alternative: using the exponential generating function:

egf := 1 + t^2*(cosh(2*sinh(t*x/2)/t));

ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):

Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, 2*(n-k+1)), k = 0..n):

seq(print(Trow(n)), n = 0..9); # Peter Luschny, Feb 29 2024

MATHEMATICA

T[n_, n_] = 1; T[n_ /; n >= 0, k_] /; 0 <= k < n := T[n, k] = T[n - 1, k - 1] + k^2*T[n - 1, k]; T[_, _] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

(* Jean-François Alcover, Nov 27 2017 *)

PROG

(SageMath) # uses[PtransMatrix from A269941]

stirset2 = lambda n: 1 if n == 1 else 1/(n*(4*n-2))

norm = lambda n, k: (-1)^k*factorial(2*n)/factorial(2*k)

M = PtransMatrix(7, stirset2, norm)

for m in M: print(m)

CROSSREFS

Columns k=0..5 give A000007, A000012, A002450(n-1), A002451(n-3), A383838(n-4), A383840(n-5).

Variants are: A008957, A036969.

Cf. A007318 (order 0), A048993 (order 1), A269948 (order 3).

Cf. A000330 (subdiagonal), A002450 (column 2), A135920 (row sums), A269941, A269944 (Stirling cycle), A298851 (central terms).