A351761
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 4, 12, 24, 40, 60, ...
0, 21, 102, 279, 588, 1065, ...
0, 148, 1160, 4332, 11536, 25220, ...
0, 1305, 16490, 84075, 282900, 746525, ...
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T(n, k) = n!*sum(j=0, n, k^(n-j)*(n-j)^j/j!);
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T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));
A351768
a(n) = n! * Sum_{k=0..n} k^(n-k) * (n-k)^k/k!.
Original entry on oeis.org
1, 0, 2, 18, 276, 6260, 190950, 7523082, 371286440, 22356290952, 1608686057610, 136069954606190, 13345029902628732, 1500054487474871484, 191349476316804534638, 27464505325501082617170, 4402551348139824475260240, 783025812197886669354545552
Offset: 0
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Join[{1}, Table[n!*Sum[k^(n-k) * (n-k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 19 2022 *)
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a(n) = n!*sum(k=0, n, k^(n-k)*(n-k)^k/k!);
A351779
a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * (n-k)^k/k!.
Original entry on oeis.org
1, -1, 4, -63, 2288, -138525, 12381084, -1528482823, 249005711296, -51739455340953, 13353206066063900, -4190486732316600771, 1571373340568392914288, -693899460077821703051125, 356404409990391961980227068, -210670220153918100996704166975
Offset: 0
A295623
a(n) = n! * [x^n] exp(n*x*exp(x)).
Original entry on oeis.org
1, 1, 8, 90, 1424, 28900, 716292, 20972098, 708317248, 27108056808, 1159375192100, 54799938951934, 2836735081572240, 159606310760007436, 9698172715195196260, 632924646574215596850, 44153807025286701187328, 3278903858941755472870864, 258247909552273997037934788
Offset: 0
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Table[n! SeriesCoefficient[Exp[n x Exp[x]], {x, 0, n}], {n, 0, 18}]
Table[Sum[BellY[n, k, n Range[n]], {k, 0, n}], {n, 0, 18}]
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a(n) = sum(k=0, n, n^k*k^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Jul 04 2022
Showing 1-4 of 4 results.