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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 n>=0 and 0<=k<=n. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Also known as central factorial numbers |t(2n, 2k)| (cf. A008955).

LINKS

Table of n, a(n) for n=0..44.

Peter Luschny, The P-transform.

FORMULA

T(n,k) = (-1)^k*((2n)!/(2k)!)*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. Compare also 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).

EXAMPLE

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]

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);

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-Fran├žois Alcover, Jul 25 2019 *)

PROG

(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

CROSSREFS

Variant: A008955.

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

Cf. A000330, A001044, A101686.

Sequence in context: A016714 A211799 A113950 * A121906 A028360 A010301

Adjacent sequences:  A269941 A269942 A269943 * A269945 A269946 A269947

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Mar 22 2016

STATUS

approved

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Last modified December 8 12:07 EST 2019. Contains 329862 sequences. (Running on oeis4.)