A382735
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))^3.
Original entry on oeis.org
1, 0, 0, 0, 3, 0, 0, 3, 3, 0, 0, 3, 27, 3, 0, 0, 3, 75, 75, 3, 0, 0, 3, 171, 579, 171, 3, 0, 0, 3, 363, 2667, 2667, 363, 3, 0, 0, 3, 747, 10083, 22779, 10083, 747, 3, 0, 0, 3, 1515, 34635, 142923, 142923, 34635, 1515, 3, 0, 0, 3, 3051, 112899, 761211, 1396803, 761211, 112899, 3051, 3, 0
Offset: 0
Square array begins:
1, 0, 0, 0, 0, 0, ...
0, 3, 3, 3, 3, 3, ...
0, 3, 27, 75, 171, 363, ...
0, 3, 75, 579, 2667, 10083, ...
0, 3, 171, 2667, 22779, 142923, ...
0, 3, 363, 10083, 142923, 1396803, ...
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a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n, j, 2)*stirling(k, j, 2));
A382737
a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling2(n,k)^2.
Original entry on oeis.org
1, 2, 14, 254, 8654, 467102, 36414734, 3862847774, 534433092494, 93409669590302, 20117959360842254, 5233190283794276894, 1617259866279958581134, 585633786711715561283102, 245587300036701328750786574, 118067003149791582488105955614, 64502003996859329263691323378574
Offset: 0
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f:= proc(n) local k; add(k!*(k+1)!*Stirling2(n,k)^2, k=0..n) end proc:
map(f, [$0..40]);
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Table[Sum[k! * (k+1)! * StirlingS2[n,k]^2, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Apr 13 2025 *)
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a(n) = sum(k=0, n, k!*(k+1)!*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
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a(n) = sum(k=0, n, k!^2*binomial(k+3, 3)*stirling(n, k, 2)^2);
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);
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.