A382673
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))^3.
Original entry on oeis.org
1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 22, 52, 22, 1, 1, 46, 208, 208, 46, 1, 1, 94, 736, 1372, 736, 94, 1, 1, 190, 2440, 7516, 7516, 2440, 190, 1, 1, 382, 7792, 37012, 60316, 37012, 7792, 382, 1, 1, 766, 24328, 170668, 418996, 418996, 170668, 24328, 766, 1, 1, 1534, 74896, 754132, 2653036, 3964684, 2653036, 754132, 74896, 1534, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 4, 10, 22, 46, 94, ...
1, 10, 52, 208, 736, 2440, ...
1, 22, 208, 1372, 7516, 37012, ...
1, 46, 736, 7516, 60316, 418996, ...
1, 94, 2440, 37012, 418996, 3964684, ...
...
-
a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2));
A382736
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))^4.
Original entry on oeis.org
1, 0, 0, 0, 4, 0, 0, 4, 4, 0, 0, 4, 44, 4, 0, 0, 4, 124, 124, 4, 0, 0, 4, 284, 1084, 284, 4, 0, 0, 4, 604, 5164, 5164, 604, 4, 0, 0, 4, 1244, 19804, 48044, 19804, 1244, 4, 0, 0, 4, 2524, 68524, 313804, 313804, 68524, 2524, 4, 0, 0, 4, 5084, 224284, 1707884, 3281404, 1707884, 224284, 5084, 4, 0
Offset: 0
Square array begins:
1, 0, 0, 0, 0, 0, ...
0, 4, 4, 4, 4, 4, ...
0, 4, 44, 124, 284, 604, ...
0, 4, 124, 1084, 5164, 19804, ...
0, 4, 284, 5164, 48044, 313804, ...
0, 4, 604, 19804, 313804, 3281404, ...
-
a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*stirling(n, j, 2)*stirling(k, j, 2));
A382678
a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+3,3) * Stirling2(n+1,k+1)^2.
Original entry on oeis.org
1, 5, 77, 2357, 118061, 8712245, 886143917, 118592620277, 20176999414061, 4249819031692085, 1084956766012858157, 329975948760472311797, 117851658189070970988461, 48830366210401091606537525, 23228207308210113849419226797, 12571433948267218576823401692917
Offset: 0
-
a(n) = sum(k=0, n, k!^2*binomial(k+3, 3)*stirling(n+1, k+1, 2)^2);
A382677
a(n) = 9 - 28 * 2^n + 20 * 3^n.
Original entry on oeis.org
1, 13, 77, 325, 1181, 3973, 12797, 40165, 124061, 379333, 1152317, 3485605, 10514141, 31657093, 95200637, 286060645, 859099421, 2579133253, 7741069757, 23230549285, 69706327901, 209148343813, 627503751677, 1882628695525, 5648120967581, 16944832664773
Offset: 0
Showing 1-4 of 4 results.