A290158
a(n) = n! * [x^n] exp(-n*x)/(1 + LambertW(-x)).
Original entry on oeis.org
1, 0, 4, -9, 208, -1525, 33516, -463099, 11293248, -231839577, 6517863100, -175791146311, 5723314711632, -189288946716181, 7083626583237036, -275649085963046475, 11724766124450058496, -522717581675749841713, 24981438186138642481404
Offset: 0
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Table[n! SeriesCoefficient[Exp[-n x]/(1 + LambertW[-x]), {x, 0, n}], {n, 0, 18}]
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a(n) = (-1)^n*n!*sum(k=0, n\2, n^k*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, May 05 2023
A356811
a(n) = Sum_{k=0..n} (k*n+1)^(n-k) * binomial(n,k).
Original entry on oeis.org
1, 2, 8, 71, 1040, 22457, 676000, 26861977, 1347932416, 82873789793, 6114540967424, 532596023373713, 53990083205042176, 6289985311473281329, 833180470332123750400, 124356049859476364116193, 20754548375601491155681280, 3847574240184742568296430273
Offset: 0
A356814
a(n) = Sum_{k=0..n} (-1)^k * (k*n+1)^(n-k) * binomial(n,k).
Original entry on oeis.org
1, 0, -4, -27, -64, 4375, 199584, 6739607, 169934848, -1012395105, -709624000000, -86599643309201, -8221227668471808, -638169258399740977, -27617164284655812608, 3853095093357099609375, 1568756883209662050074624, 360407172063462944082773311
Offset: 0
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a(n) = sum(k=0, n, (-1)^k*(k*n+1)^(n-k)*binomial(n, k));
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a(n) = n!*sum(k=0, n\2, (-1)^k*n^(n-k)*stirling(n-k, k, 2)/(n-k)!);
A356817
a(n) = Sum_{k=0..n} (-1)^k * (k*n-1)^(n-k) * binomial(n,k).
Original entry on oeis.org
1, -2, 0, 1, 144, 4143, 110368, 2535475, 13299968, -5169863825, -639341093376, -59073970497885, -4677854594527232, -276406098219258425, 2399871442122924032, 5163244810691492730907, 1331213942683118587674624, 262517264591996332314037215
Offset: 0
A362839
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling2(n-j,j)/(n-j)!.
Original entry on oeis.org
1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 16, 0, 1, 0, 8, 27, 80, 65, 0, 1, 0, 10, 48, 216, 560, 336, 0, 1, 0, 12, 75, 448, 2025, 4512, 1897, 0, 1, 0, 14, 108, 800, 5120, 21708, 40768, 11824, 0, 1, 0, 16, 147, 1296, 10625, 67584, 260253, 407808, 80145, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 0, 0, 0, 0, 0, ...
0, 2, 4, 6, 8, 10, ...
0, 3, 12, 27, 48, 75, ...
0, 16, 80, 216, 448, 800, ...
0, 65, 560, 2025, 5120, 10625, ...
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T(n, k) = n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 2)/(n-j)!);
Showing 1-5 of 5 results.
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