A269944 Triangle read by rows, Stirling cycle 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) + (n-1)^2*T(n-1, k), for 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 4, 5, 1, 0, 36, 49, 14, 1, 0, 576, 820, 273, 30, 1, 0, 14400, 21076, 7645, 1023, 55, 1, 0, 518400, 773136, 296296, 44473, 3003, 91, 1, 0, 25401600, 38402064, 15291640, 2475473, 191620, 7462, 140, 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. A008955).

The analog for the Stirling set numbers is A269945.

Examples

			Triangle starts:
  [1]
  [0,     1]
  [0,     1,     1]
  [0,     4,     5,    1]
  [0,    36,    49,   14,    1]
  [0,   576,   820,  273,   30,  1]
  [0, 14400, 21076, 7645, 1023, 55, 1]
		

Links

• Peter Luschny, The P-transform.
• Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.
• B. K. Miceli, Two q-Analogues of Poly-Stirling Numbers, J. Integer Seq., 14 (2011), 11.9.6.

Crossrefs

Variants: A204579 (signed, row 0 missing), A008955.

Cf. A007318 (order 0), A132393 (order 1), A269947 (order 3).

Cf. A000330 (subdiagonal), A001044 (column 1), A101686 (row sums), A269945 (Stirling set), A269941 (P-transform).

Programs

Maple

T := proc(n, k) option remember; if n=k then return 1 fi; if k<0 or k>n then return 0 fi; T(n-1, k-1)+(n-1)^2*T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A269944_row := n -> PTrans(n, n->`if`(n=1, 1, (n-1)^2/(n*(4*n-2))), (n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269944_row(n)), n=0..8);
# From Peter Luschny, Feb 29 2024: (Start)
# Computed as the coefficients of polynomials:
P := (x, n) -> local j; mul((j - x)*(j + x), j = 0..n-1):
T := (n, k) -> (-1)^k*coeff(P(x, n), x, 2*k):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
# Alternative, using the exponential generating function:
egf := cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, n-k), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # (End)
# Alternative, row polynomials:
rowpoly := n -> pochhammer(-sqrt(x), n) * pochhammer(sqrt(x), n):
row := n -> local k; seq((-1)^k*coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..6);  # Peter Luschny, Aug 03 2024

Mathematica

T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + (n - 1)^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Jul 25 2019 *)

Sage

stircycle2 = lambda n: 1 if n == 1 else (n-1)^2/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stircycle2, 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) = (n - 1)^2 / (n*(4*n - 2)). The P-transform is defined in the link. See the Sage and Maple implementations below.

T(n, 1) = ((n - 1)!)^2 for n >= 1 (cf. A001044).

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

Row sums: Product_{k=1..n} ((k - 1)^2 + 1) for n >= 0 (cf. A101686).

From Fabián Pereyra, Apr 25 2022: (Start)

T(n,k) = (-1)^(n-k)*Sum_{j=2*k..2*n} Stirling1(2*n,j)*binomial(j,2*k)*(n-1)^(j-2*k).

T(n,k) = Sum_{j=0..2*k} (-1)^(j - k)*Stirling1(n, j)*Stirling1(n, 2*k - j). (End)

From Peter Luschny, Feb 29 2024: (Start)

T(n, k) = (-1)^k*[x^(2*k)] P(x, n) where P(x, n) = Product_{j=0..n-1} (j-x)*(j+x).

T(n, k) = (2*n)!*[t^(n-k)] [x^(2*n)] cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)). (End)

T(n, k) = (-1)^k*[x^k] Pochhammer(-sqrt(x), n) * Pochhammer(sqrt(x), n). - Peter Luschny, Aug 03 2024