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));
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));
A382674
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^4.
Original entry on oeis.org
1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 77, 29, 1, 1, 61, 325, 325, 61, 1, 1, 125, 1181, 2357, 1181, 125, 1, 1, 253, 3973, 13621, 13621, 3973, 253, 1, 1, 509, 12797, 69269, 118061, 69269, 12797, 509, 1, 1, 1021, 40165, 326005, 862261, 862261, 326005, 40165, 1021, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 5, 13, 29, 61, 125, ...
1, 13, 77, 325, 1181, 3973, ...
1, 29, 325, 2357, 13621, 69269, ...
1, 61, 1181, 13621, 118061, 862261, ...
1, 125, 3973, 69269, 862261, 8712245, ...
...
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a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 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);
A382742
Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/4) * (1 / (exp(x) + exp(y) - exp(x+y))^4 - 1).
Original entry on oeis.org
1, 1, 1, 1, 11, 1, 1, 31, 31, 1, 1, 71, 271, 71, 1, 1, 151, 1291, 1291, 151, 1, 1, 311, 4951, 12011, 4951, 311, 1, 1, 631, 17131, 78451, 78451, 17131, 631, 1, 1, 1271, 56071, 426971, 820351, 426971, 56071, 1271, 1, 1, 2551, 177691, 2093491, 6709651, 6709651, 2093491, 177691, 2551, 1
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 11, 31, 71, 151, 311, ...
1, 31, 271, 1291, 4951, 17131, ...
1, 71, 1291, 12011, 78451, 426971, ...
1, 151, 4951, 78451, 820351, 6709651, ...
1, 311, 17131, 426971, 6709651, 79008011, ...
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a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*stirling(n, j, 2)*stirling(k, j, 2))/4;
Showing 1-5 of 5 results.