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

0,5

Comments

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

Links

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

Formula

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

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

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

Example

[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,

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 *)

Prog

(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))

Crossrefs

Cf. A001662, A001147, A002866.

Sequence in context: A136160 A268439 A120362 * A010102 A326477 A285650

Adjacent sequences: A201633 A201634 A201635 * A201637 A201638 A201639

Keyword

nonn,tabl

Author

Peter Luschny, Nov 13 2012

Status

approved