A325138
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} binomial(j+k, k)*|Stirling1(n, j+k)|*(k+1)^j.
Original entry on oeis.org
1, 2, 8, 45, 320, 2730, 27054, 304584, 3832688, 53233272, 808045560, 13297113720, 235635543912, 4471304008704, 90415029604704, 1940195561267880, 44021278940004480, 1052672670160355520, 26454200168941936704, 696874344218429604480, 19198703924579071278720
Offset: 0
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a(n) = sum(k=0, n, sum(j=0, n-k, binomial(j+k, k)*abs(stirling(n, j+k, 1))*(k+1)^j)); \\ Michel Marcus, Apr 13 2019
A325139
Triangle T(n, k) = [t^n] Gamma(n + k + m + t)/Gamma(k + m + t) for m = 2 and 0 <= k <= n, read by rows.
Original entry on oeis.org
1, 2, 1, 6, 7, 1, 24, 47, 15, 1, 120, 342, 179, 26, 1, 720, 2754, 2070, 485, 40, 1, 5040, 24552, 24574, 8175, 1075, 57, 1, 40320, 241128, 305956, 134449, 24885, 2086, 77, 1, 362880, 2592720, 4028156, 2231012, 541849, 63504, 3682, 100, 1
Offset: 0
0: 1;
1: 2, 1;
2: 6, 7, 1;
3: 24, 47, 15, 1;
4: 120, 342, 179, 26, 1;
5: 720, 2754, 2070, 485, 40, 1;
6: 5040, 24552, 24574, 8175, 1075, 57, 1;
7: 40320, 241128, 305956, 134449, 24885, 2086, 77, 1;
8: 362880, 2592720, 4028156, 2231012, 541849, 63504, 3682, 100, 1;
9: 3628800, 30334320, 56231712, 37972304, 11563650, 1768809, 142632, 6054, 126, 1;
A: A000142, A001711, A001717, A001723, ...
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T := (n, k) -> add(binomial(j+k, k)*(k+2)^j*abs(Stirling1(n, j+k)), j=0..n-k):
seq(seq(T(n, k), k=0..n), n=0..8);
# Note that for n > 16 Maple fails (at least in some versions) to compute the
# terms properly. Inserting 'simplify' or numerical evaluation might help.
A325139Row := proc(n) local ogf, ser; ogf := (n, k) -> GAMMA(n+k+2+x)/GAMMA(k+2+x);
ser := (n, k) -> series(ogf(n,k), x, k+2); seq(coeff(ser(n,k), x, k), k=0..n) end:
seq(A325139Row(n), n=0..9);
Showing 1-2 of 2 results.