A349470 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,n).
1, 1, 5, 65, 1394, 40378, 1470972, 64575585, 3315911300, 194921240846, 12905391110105, 950172113032181, 77000666619646717, 6810514097879311450, 652810277600420281734, 67407087759052608218945, 7459157975936646185855880, 880616251774021869817185430
Offset: 0
Keywords
Examples
a(1) = (1/1!) * (1) = 1. a(2) = (1/2!) * (-1*2 + 3*4) = 5. a(3) = (1/3!) * (1*2*3 - 4*5*6 + 7*8*9) = 65. a(4) = (1/4!) * (-1*2*3*4 + 5*6*7*8 - 9*10*11*12 + 13*14*15*16) = 1394.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..338
Programs
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Mathematica
a[n_] := Sum[(-1)^(n - k) * Binomial[k*n, n], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2021 *) -
PARI
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k*n, n));
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PARI
a(n) = sum(j=0, n, (-1)^(n-j)*prod(k=(j-1)*n+1, j*n, k))/n!;
Formula
a(n) = (1/n!) * Sum_{j=0..n} (-1)^(n-j) * Product_{k=(j-1)*n+1..j*n} k.
a(n) ~ exp(n + 1/2) * n^(n - 1/2) / (sqrt(2*Pi) * (1 + exp(1))). - Vaclav Kotesovec, Nov 20 2021