A277372
a(n) = Sum_{k=1..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!.
Original entry on oeis.org
0, 1, 10, 141, 2584, 58745, 1602576, 51165205, 1874935168, 77644293201, 3588075308800, 183111507687581, 10230243235200000, 621111794820235849, 40722033570202507264, 2867494972696071121125, 215840579093024990396416, 17294837586403146090259745, 1469799445329208661211021312
Offset: 0
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a := n -> add(binomial(n,n-k)*n^(n-k)*n!/(n-k)!, k=1..n):
seq(a(n), n=0..18);
# Alternatively:
A277372 := n -> n!*LaguerreL(n,-n) - n^n:
seq(simplify(A277372(n)), n=0..18);
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a(n) = sum(k=1, n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!); \\ Michel Marcus, Oct 12 2016
A329655
Square array read by antidiagonals: T(n,k) is the number of relations between set A with n elements and set B with k elements that are both right unique and left unique.
Original entry on oeis.org
1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 33, 20, 5, 6, 30, 72, 72, 30, 6, 7, 42, 135, 208, 135, 42, 7, 8, 56, 228, 500, 500, 228, 56, 8, 9, 72, 357, 1044, 1545, 1044, 357, 72, 9, 10, 90, 528, 1960, 4050, 4050, 1960, 528, 90, 10, 11, 110, 747, 3392, 9275, 13326, 9275, 3392, 747, 110, 11
Offset: 1
The symmetric array T(n,k) begins:
1, 2, 3, 4, 5, 6, 7, 8, 9, ...
2, 6, 12, 20, 30, 42, 56, 72, 90, ...
3, 12, 33, 72, 135, 228, 357, 528, 747, ...
4, 20, 72, 208, 500, 1044, 1960, 3392, 5508, ...
5, 30, 135, 500, 1545, 4050, 9275, 19080, 36045, ...
6, 42, 228, 1044, 4050, 13326, 37632, 93288, 207774, ...
7, 56, 357, 1960, 9275, 37632, 130921, 394352, 1047375, ...
8, 72, 528, 3392, 19080, 93288, 394352, 1441728, 4596552, ...
9, 90, 747, 5508, 36045, 207774, 1047375, 4596552, 17572113, ...
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T:= (n,k)-> value(Sum(binomial(n,j)*binomial(k, j)*j!, j=1..k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);
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T[n_, k_] := Sum[Binomial[n, j] * Binomial[k, j] * j!, {j, 1, k}]; Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)
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T:=(n,k)->_plus (binomial(n,j)*binomial(k, j)* j! $ j=1..k):
Showing 1-2 of 2 results.
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