A382734
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (exp(x) + exp(y) - exp(x+y))^2.
Original entry on oeis.org
1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 14, 2, 0, 0, 2, 38, 38, 2, 0, 0, 2, 86, 254, 86, 2, 0, 0, 2, 182, 1118, 1118, 182, 2, 0, 0, 2, 374, 4142, 8654, 4142, 374, 2, 0, 0, 2, 758, 14078, 51662, 51662, 14078, 758, 2, 0, 0, 2, 1526, 45614, 267566, 467102, 267566, 45614, 1526, 2, 0
Offset: 0
Square array begins:
1, 0, 0, 0, 0, 0, ...
0, 2, 2, 2, 2, 2, ...
0, 2, 14, 38, 86, 182, ...
0, 2, 38, 254, 1118, 4142, ...
0, 2, 86, 1118, 8654, 51662, ...
0, 2, 182, 4142, 51662, 467102, ...
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a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*stirling(n, j, 2)*stirling(k, j, 2));
A382738
a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling2(n,k)^2.
Original entry on oeis.org
1, 3, 27, 579, 22779, 1396803, 121998267, 14333812419, 2175860165499, 414000255441603, 96422983358827707, 26970211126038920259, 8918364340126714711419, 3440770498298077165166403, 1531504734740033368269820347, 778873986278207207346380124099
Offset: 0
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Table[Sum[k! * (k+2)! * StirlingS2[n,k]^2/2, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 30 2025 *)
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a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n, k, 2)^2);
A382739
a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+3,3) * Stirling2(n,k)^2.
Original entry on oeis.org
1, 4, 44, 1084, 48044, 3281404, 316032044, 40592233084, 6687195379244, 1372291071723004, 342877475325619244, 102409872018962876284, 36014541870868393113644, 14724003012156426011095804, 6922777830859189006847193644, 3708347961746448904830944962684
Offset: 0
-
a(n) = sum(k=0, n, k!^2*binomial(k+3, 3)*stirling(n, k, 2)^2);
A382804
a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling1(n,k)^2.
Original entry on oeis.org
1, 2, 14, 260, 9588, 581952, 52096512, 6423520896, 1041005447424, 214260350714496, 54547409318781312, 16820040059243046144, 6175245603727007034624, 2661063379044058584861696, 1329787781176741647226481664, 762665713456216694195942866944
Offset: 0
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a(n) = sum(k=0, n, k!*(k+1)!*stirling(n, k, 1)^2);
A382847
a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (Stirling2(n,k) * k!)^2.
Original entry on oeis.org
1, 1, 14, 579, 48044, 6647405, 1379024730, 400315753159, 154879704709784, 77018569697097009, 47863427797633958630, 36348262891572161261963, 33119479438137288670256964, 35660343372397246917403353013, 44791475616825872944740798413234, 64911462519379469821754507087299215
Offset: 0
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Table[Sum[Binomial[n + k - 1, k] (StirlingS2[n, k] k!)^2, {k, 0, n}], {n, 0, 15}]
Table[(n!)^2 SeriesCoefficient[1/(Exp[x] + Exp[y] - Exp[x + y])^n, {x, 0, n}, {y, 0, n}], {n, 0, 15}]
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a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 2))^2); \\ Seiichi Manyama, Apr 06 2025
Showing 1-5 of 5 results.