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

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..338

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

Example

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.

Mathematica

a[n_] := Sum[(-1)^(n - k) * Binomial[k*n, n], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k*n, n));
(PARI) a(n) = sum(j=0, n, (-1)^(n-j)*prod(k=(j-1)*n+1, j*n, k))/n!;

Crossrefs

Cf. A096130, A096131, A349471, A349480.

Sequence in context: A079482 A147625 A157097 * A234295 A370261 A251575

Adjacent sequences: A349467 A349468 A349469 * A349471 A349472 A349473

Keyword

nonn

Author

Seiichi Manyama, Nov 19 2021

Status

approved