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URL: https://oeis.org/A201636

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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
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.
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
(SageMath)
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 13 2012
STATUS
approved