A201636 Triangle read by rows, n>=0, k>=0, T(0,0) = 1, T(n,k) = Sum_{j=0..k} (C(n+k,k-j)*(-1)^(k-j)*2^(n-j)*Sum_{i=0..j} (C(n+j,i)*|S(n+j-i,j-i)|)), S = Stirling number of first kind.

1, 0, 1, 0, 4, 3, 0, 24, 40, 15, 0, 192, 520, 420, 105, 0, 1920, 7392, 9520, 5040, 945, 0, 23040, 116928, 211456, 176400, 69300, 10395, 0, 322560, 2055168, 4858560, 5642560, 3465000, 1081080, 135135, 0, 5160960, 39896064, 117722880, 177580480, 150870720, 73153080, 18918900, 2027025

Offset: 0

Author

Peter Luschny, Nov 13 2012

Keywords

• nonn
• tabl

Comments

This triangle was inspired by a formula of Vladimir Kruchinin given in A001662.

Examples

			[n\k 0,    1,    2,    3,    4,   5]
[0]  1,
[1]  0,    1,
[2]  0,    4,    3,
[3]  0,   24,   40,   15,
[4]  0,  192,  520,  420,  105,
[5]  0, 1920, 7392, 9520, 5040, 945,
		

Crossrefs

Cf. A001662, A001147, A002866.

Programs

Maple

A201636 := proc(n,k) if n=0 and k=0 then 1 else
add(binomial(n+k,k-j)*(-1)^(n+k-j)*2^(n-j)*
add(binomial(n+j,i)*stirling1(n+j-i,j-i),i=0..j),j=0..k) fi end:
for n from 0 to 8 do print(seq(A201636(n,k),k=0..n)) od;

Mathematica

T[0, 0] = 1; T[n_, k_] := T[n, k] = Sum[Binomial[n+k, k-j]*(-1)^(n+k-j)* 2^(n-j)*Sum[Binomial[n+j, i]*StirlingS1[n+j-i, j-i], {i, 0, j}], {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-François Alcover, Jun 29 2019 *)

Sage

def A201636(n,k) :
    if n==0 and k==0: return 1
    return add(binomial(n+k,k-j)*(-1)^(k-j)*2^(n-j)*add(binomial(n+j,i)* stirling_number1(n+j-i,j-i) for i in (0..j)) for j in (0..k))

Formula

T(n,1) = A002866(n) for n>0.

T(n,n) = A001147(n).

Sum((-1)^(n-k)*T(n,k)) = A001662(n+1).