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));
A382738
a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling2(n,k)^2.
Original entry on oeis.org
1, 3, 27, 579, 22779, 1396803, 121998267, 14333812419, 2175860165499, 414000255441603, 96422983358827707, 26970211126038920259, 8918364340126714711419, 3440770498298077165166403, 1531504734740033368269820347, 778873986278207207346380124099
Offset: 0
-
Table[Sum[k! * (k+2)! * StirlingS2[n,k]^2/2, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 30 2025 *)
-
a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n, k, 2)^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);
A382828
a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling1(n+1,k+1)^2.
Original entry on oeis.org
1, 4, 55, 1623, 82116, 6302028, 680105112, 98011315608, 18163969766592, 4205977241171328, 1189459906531372224, 403300593144673493184, 161454763431242385682176, 75337361633768810384542464, 40524573487904551618353921024, 24890567631479746511661428751360
Offset: 0
-
a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n+1, k+1, 1)^2);
Showing 1-4 of 4 results.