A268438 Triangle read by rows, T(n,k) = (-1)^k*(2*n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

1, 0, 1, 0, 8, 6, 0, 180, 240, 90, 0, 8064, 14560, 10080, 2520, 0, 604800, 1330560, 1285200, 604800, 113400, 0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400, 0, 10897286400, 30858347520, 43770767040, 36970053120, 18918900000, 5448643200, 681080400

Offset: 0

Author

Peter Luschny, Mar 07 2016

Keywords

• nonn
• tabl

Comments

The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

Examples

			Triangle starts:
[1],
[0, 1],
[0, 8,        6],
[0, 180,      240,       90],
[0, 8064,     14560,     10080,     2520],
[0, 604800,   1330560,   1285200,   604800,    113400],
[0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400].
		

Links

• Peter Luschny, The P-transform.
• Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.

Crossrefs

Cf. A000680, A060593, A268437, A269940, A269941.

Programs

Maple

A268438 := proc(n,k) local F,T;
  F := proc(n,k) option remember;
  `if`(n=0 and k=0, 1,`if`(n=k, (4*n-2)*F(n-1,k-1),
  F(n-1,k)*(n+k))) end;
  T := proc(n, k) option remember;
  `if`(k=0 and n=0, 1,`if`(k<=0 or k>n, 0,
  (4*n-2)*n*(n+k-1)*(T(n-1,k)+T(n-1,k-1)))) end:
T(n,k)/F(n,k) end:
for n from 0 to 6 do seq(A268438(n,k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268438_row := n -> PTrans(n, n->n/(n+1),(n,k)->(-1)^k*(2*n)!):
seq(lprint(A268438_row(n)), n=0..8);

Mathematica

T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)

Sage

A268438 = lambda n,k: (factorial(2*n)/falling_factorial(n+k,n))*sum((-1)^(m+k)* binomial(n+k,n+m)*stirling_number1(n+m,m) for m in (0..k))
for n in (0..7): print([A268438(n,m) for m in (0..n)])

Sage

# uses[PtransMatrix from A269941]
PtransMatrix(7, lambda n: n/(n+1), lambda n,k: (-1)^k*factorial(2*n))

Formula

T(n,k) = ((2*n)!/FF(n+k,n))*Sum_{m=0..k}(-1)^(m+k)*C(n+k,n+m)*Stirling1(n+m,m) where FF denotes the falling factorial function.

T(n,k) = ((2*n)!/FF(n+k,n))*A269940(n,k).

T(n,1) = (2*n)!/(n+1) = A060593(n) for n>=1.

T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.