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, ...
-
a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*stirling(n, j, 2)*stirling(k, j, 2));
A382741
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 / (exp(x) + exp(y) - exp(x+y))^3 - 1).
Original entry on oeis.org
1, 1, 1, 1, 9, 1, 1, 25, 25, 1, 1, 57, 193, 57, 1, 1, 121, 889, 889, 121, 1, 1, 249, 3361, 7593, 3361, 249, 1, 1, 505, 11545, 47641, 47641, 11545, 505, 1, 1, 1017, 37633, 253737, 465601, 253737, 37633, 1017, 1, 1, 2041, 118969, 1228249, 3657721, 3657721, 1228249, 118969, 2041, 1
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 9, 25, 57, 121, 249, ...
1, 25, 193, 889, 3361, 11545, ...
1, 57, 889, 7593, 47641, 253737, ...
1, 121, 3361, 47641, 465601, 3657721, ...
1, 249, 11545, 253737, 3657721, 40666089, ...
-
a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n, j, 2)*stirling(k, j, 2))/3;
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, ...
-
a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*stirling(n, j, 2)*stirling(k, j, 2))/4;
A382801
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 / (1 - log(1-x) * log(1-y))^2 - 1).
Original entry on oeis.org
1, 1, 1, 2, 7, 2, 6, 20, 20, 6, 24, 72, 130, 72, 24, 120, 324, 642, 642, 324, 120, 720, 1764, 3468, 4794, 3468, 1764, 720, 5040, 11304, 21372, 32964, 32964, 21372, 11304, 5040, 40320, 83448, 150120, 238404, 290976, 238404, 150120, 83448, 40320
Offset: 1
Square array begins:
1, 1, 2, 6, 24, 120, ...
1, 7, 20, 72, 324, 1764, ...
2, 20, 130, 642, 3468, 21372, ...
6, 72, 642, 4794, 32964, 238404, ...
24, 324, 3468, 32964, 290976, 2524080, ...
120, 1764, 21372, 238404, 2524080, 26048256, ...
-
a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*abs(stirling(n, j, 1)*stirling(k, j, 1)))/2;
Showing 1-4 of 4 results.