A349470 - OEIS

%I #36 Nov 20 2021 08:01:59

%S 1,1,5,65,1394,40378,1470972,64575585,3315911300,194921240846,

%T 12905391110105,950172113032181,77000666619646717,6810514097879311450,

%U 652810277600420281734,67407087759052608218945,7459157975936646185855880,880616251774021869817185430

%N a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,n).

%H Seiichi Manyama, <a href="/A349470/b349470.txt">Table of n, a(n) for n = 0..338</a>

%F a(n) = (1/n!) * Sum_{j=0..n} (-1)^(n-j) * Product_{k=(j-1)*n+1..j*n} k.

%F a(n) ~ exp(n + 1/2) * n^(n - 1/2) / (sqrt(2*Pi) * (1 + exp(1))). - _Vaclav Kotesovec_, Nov 20 2021

%e a(1) = (1/1!) * (1) = 1.

%e a(2) = (1/2!) * (-1*2 + 3*4) = 5.

%e a(3) = (1/3!) * (1*2*3 - 4*5*6 + 7*8*9) = 65.

%e a(4) = (1/4!) * (-1*2*3*4 + 5*6*7*8 - 9*10*11*12 + 13*14*15*16) = 1394.

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

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k*n, n));

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

%Y Cf. A096130, A096131, A349471, A349480.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 19 2021