A202185 Triangle T(n,m) = coefficient of x^n in expansion of x^m*(x+1)^(log(1+x)*m) = sum(n>=m, T(n,m) x^n*m!/n!).

1, 0, 1, 6, 0, 1, -24, 24, 0, 1, 170, -120, 60, 0, 1, -1320, 1380, -360, 120, 0, 1, 11816, -14280, 6090, -840, 210, 0, 1, -118944, 171808, -77280, 19600, -1680, 336, 0, 1, 1329156, -2249856, 1181376, -292320, 51660, -3024, 504, 0, 1, -16313760, 32093280

Offset

1,4

Comments

Also the Bell transform of (-1)^n*(n+1)*Sum_{k=0..n} S1(n,2*k)*(2*k)!/k! where S1 are the Stirling cycle numbers A132393. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 15 2016

Links

Indranil Ghosh, Table of n, a(n) for n = Rows 1..50, flattened

Formula

T(n,m) = binomial(n,m)*sum(k=0..n-m, ((2*k)!*m^k*stirling1(n-m,2*k))/k!).

Example

1,
0, 1,
6, 0, 1,
-24, 24, 0, 1,
170, -120, 60, 0, 1,
-1320, 1380, -360, 120, 0, 1,
11816, -14280, 6090, -840, 210, 0, 1

Mathematica

Flatten[Table[Binomial[n, m]*Sum[((2k)!*m^k*StirlingS1[n-m, 2k])/k!, {k, 0, n-m}], {n, 1, 7}, {m, 1, n}]] (* Indranil Ghosh, Feb 21 2017 *)

Prog

(Maxima)
T(n, m):=binomial(n, m)*sum(((2*k)!*m^k*stirling1(n-m, 2*k))/k!, k, 0, n-m);
(Sage) # uses[bell_transform from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
def A202185_row(n):
    f = lambda n: (-1)^n*(n+1)*sum(factorial(2*k)*stirling_number1(n, 2*k)/ factorial(k) for k in (0..n))
    return bell_transform(n, [f(k) for k in (0..n)])
[A202185_row(n) for n in (0..9)] # Peter Luschny, Jan 15 2016
(PARI) T(n, m) = binomial(n, m)*sum(k=0, n-m, ((2*k)!*m^k*stirling(n-m, 2*k, 1))/k!); \\ Michel Marcus, Jan 16 2016

Crossrefs

Sequence in context: A227612 A221273 A352607 * A304334 A303535 A269340

Adjacent sequences: A202182 A202183 A202184 * A202186 A202187 A202188

Keyword

sign,tabl

Author

Vladimir Kruchinin, Dec 13 2011

Status

approved