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

Offset: 0

Author

Peter Luschny, Mar 22 2016

Keywords

• nonn
• tabl

Comments

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

The analog for the Stirling cycle numbers is A269944.

Examples

			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]
		

Links

• 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.

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).

Programs

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;
# Alternatively 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);
# 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 *)

Sage

# 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)

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