A382806
a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling1(n,k)^2.
Original entry on oeis.org
1, 3, 27, 588, 24612, 1669128, 165049224, 22269896064, 3918921022656, 870149951146944, 237662482188210624, 78249086559726140160, 30547324837444471084800, 13946361918619108837939200, 7359961832428044552536217600, 4444946383758589481684168540160
Offset: 0
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a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n, k, 1)^2);
A382799
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (1 - log(1-x) * log(1-y))^2.
Original entry on oeis.org
1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 14, 4, 0, 0, 12, 40, 40, 12, 0, 0, 48, 144, 260, 144, 48, 0, 0, 240, 648, 1284, 1284, 648, 240, 0, 0, 1440, 3528, 6936, 9588, 6936, 3528, 1440, 0, 0, 10080, 22608, 42744, 65928, 65928, 42744, 22608, 10080, 0, 0, 80640, 166896, 300240, 476808, 581952, 476808, 300240, 166896, 80640, 0
Offset: 0
Square array begins:
1, 0, 0, 0, 0, 0, ...
0, 2, 2, 4, 12, 48, ...
0, 2, 14, 40, 144, 648, ...
0, 4, 40, 260, 1284, 6936, ...
0, 12, 144, 1284, 9588, 65928, ...
0, 48, 648, 6936, 65928, 581952, ...
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a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*abs(stirling(n, j, 1)*stirling(k, j, 1)));
A382827
a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling1(n+1,k+1)^2.
Original entry on oeis.org
1, 3, 34, 854, 37556, 2546852, 246113904, 32104625520, 5433891955968, 1157778241057152, 303197684900579712, 95717977509042032256, 35847800701044816248064, 15713483696924130220098816, 7969364997624587289470810112, 4630203661005094483980386924544
Offset: 0
-
a(n) = sum(k=0, n, k!*(k+1)!*stirling(n+1, k+1, 1)^2);
A382853
a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (k! * Stirling1(n,k))^2.
Original entry on oeis.org
1, 1, 14, 588, 51064, 7542780, 1688795184, 532244030976, 224335607135616, 121793234373123840, 82750681453274478720, 68773648886955417943296, 68628724852793337500166144, 80970628401965472953705395200, 111490683570184861858636405923840, 177177650274516448010905794637332480
Offset: 0
-
Table[Sum[Binomial[n+k-1,k] * k!^2 * StirlingS1[n,k]^2, {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 07 2025 *)
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a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 1))^2);
Showing 1-4 of 4 results.