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A268437
Triangle read by rows, T(n,k) = (-1)^k*(2*n)!*P[n,k](1/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.
7
1, 0, 1, 0, 4, 6, 0, 30, 120, 90, 0, 336, 2800, 5040, 2520, 0, 5040, 80640, 264600, 302400, 113400, 0, 95040, 2827440, 15190560, 29937600, 24948000, 7484400, 0, 2162160, 118198080, 983782800, 2986663680, 4162158000, 2724321600, 681080400
OFFSET
0,5
COMMENTS
The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
FORMULA
T(n,k) = ((2*n)!/FF(n+k,n))*Sum_{m=0..k}(-1)^(m+k)*C(n+k,n+m)*Stirling2(n+m,m) where FF denotes the falling factorial function.
T(n,k) = ((2*n)!/FF(n+k,n))*A269939(n,k).
T(n,1) = (2*n)!/(n+1)! = A001761(n) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.
EXAMPLE
[1],
[0, 1],
[0, 4, 6],
[0, 30, 120, 90],
[0, 336, 2800, 5040, 2520],
[0, 5040, 80640, 264600, 302400, 113400],
[0, 95040, 2827440, 15190560, 29937600, 24948000, 7484400].
MAPLE
A268437 := 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*(k*T(n-1, k)+(n+k-1)*T(n-1, k-1)))) end;
T(n, k)/F(n, k) end:
for n from 0 to 6 do seq(A268437(n, k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268437_row := n -> PTrans(n, n->1/(n+1), (n, k)->(-1)^k*(2*n)!):
seq(print(A268437_row(n)), n=0..8);
MATHEMATICA
T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] StirlingS2[n+m, m], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)
PROG
(Sage)
A268437 = lambda n, k: (factorial(2*n)/falling_factorial(n+k, n))*sum((-1)^(m+k)* binomial(n+k, n+m)*stirling_number2(n+m, m) for m in (0..k))
for n in (0..7): print([A268437(n, m) for m in (0..n)])
(Sage) # uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: 1/(n+1), lambda n, k: (-1)^k* factorial(2*n))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 07 2016
STATUS
approved