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));
A382740
Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).
Original entry on oeis.org
1, 1, 1, 1, 7, 1, 1, 19, 19, 1, 1, 43, 127, 43, 1, 1, 91, 559, 559, 91, 1, 1, 187, 2071, 4327, 2071, 187, 1, 1, 379, 7039, 25831, 25831, 7039, 379, 1, 1, 763, 22807, 133783, 233551, 133783, 22807, 763, 1, 1, 1531, 71839, 636679, 1748791, 1748791, 636679, 71839, 1531, 1
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 7, 19, 43, 91, 187, ...
1, 19, 127, 559, 2071, 7039, ...
1, 43, 559, 4327, 25831, 133783, ...
1, 91, 2071, 25831, 233551, 1748791, ...
1, 187, 7039, 133783, 1748791, 18207367, ...
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a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*stirling(n, j, 2)*stirling(k, j, 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;
A382802
Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/3) * (1 / (1 - log(1-x) * log(1-y))^3 - 1).
Original entry on oeis.org
1, 1, 1, 2, 9, 2, 6, 26, 26, 6, 24, 94, 196, 94, 24, 120, 424, 996, 996, 424, 120, 720, 2312, 5448, 8204, 5448, 2312, 720, 5040, 14832, 33816, 58544, 58544, 33816, 14832, 5040, 40320, 109584, 238656, 431632, 556376, 431632, 238656, 109584, 40320
Offset: 1
Square array begins:
1, 1, 2, 6, 24, 120, ...
1, 9, 26, 94, 424, 2312, ...
2, 26, 196, 996, 5448, 33816, ...
6, 94, 996, 8204, 58544, 431632, ...
24, 424, 5448, 58544, 556376, 5017480, ...
120, 2312, 33816, 431632, 5017480, 55016408, ...
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a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*abs(stirling(n, j, 1)*stirling(k, j, 1)))/3;
Showing 1-4 of 4 results.