Search: a356817 -id:a356817
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A356806
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a(n) = Sum_{k=0..n} (k*n-1)^(n-k) * binomial(n,k).
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+10
6
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1, 0, 4, 27, 448, 10625, 344736, 14437213, 753991680, 47974773393, 3650824000000, 326917384798301, 33956137832546304, 4041303651931462969, 545552768347831566336, 82828479894303251953125, 14040577418634835164921856, 2640293357854435329683551265
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = n! * [x^n] exp( x * (exp(n * x) - 1) ).
a(n) = n! * Sum_{k=0..floor(n/2)} n^(n-k) * Stirling2(n-k,k)/(n-k)!.
a(n) = [x^n] Sum_{k>=0} x^k / (1 - (n*k-1)*x)^(k+1).
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PROG
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(PARI) a(n) = sum(k=0, n, (k*n-1)^(n-k)*binomial(n, k));
(PARI) a(n) = n!*sum(k=0, n\2, n^(n-k)*stirling(n-k, k, 2)/(n-k)!);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A356811
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a(n) = Sum_{k=0..n} (k*n+1)^(n-k) * binomial(n,k).
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+10
4
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1, 2, 8, 71, 1040, 22457, 676000, 26861977, 1347932416, 82873789793, 6114540967424, 532596023373713, 53990083205042176, 6289985311473281329, 833180470332123750400, 124356049859476364116193, 20754548375601491155681280, 3847574240184742568296430273
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = n! * [x^n] exp( x * (exp(n * x) + 1) ).
a(n) = [x^n] Sum_{k>=0} x^k / (1 - (n*k+1)*x)^(k+1).
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PROG
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(PARI) a(n) = sum(k=0, n, (k*n+1)^(n-k)*binomial(n, k));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A356814
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a(n) = Sum_{k=0..n} (-1)^k * (k*n+1)^(n-k) * binomial(n,k).
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+10
3
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1, 0, -4, -27, -64, 4375, 199584, 6739607, 169934848, -1012395105, -709624000000, -86599643309201, -8221227668471808, -638169258399740977, -27617164284655812608, 3853095093357099609375, 1568756883209662050074624, 360407172063462944082773311
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = n! * [x^n] exp( x * (1 - exp(n * x)) ).
a(n) = [x^n] Sum_{k>=0} (-x)^k / (1 - (n*k+1)*x)^(k+1).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * n^(n-k) * Stirling2(n-k,k)/(n-k)!.
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^k*(k*n+1)^(n-k)*binomial(n, k));
(PARI) a(n) = n!*sum(k=0, n\2, (-1)^k*n^(n-k)*stirling(n-k, k, 2)/(n-k)!);
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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