A358495 a(n) = Sum_{k=0..n} binomial(binomial(n, k), n).

1, 2, 1, 2, 17, 506, 48772, 13681602, 12287555282, 33669343492094, 311704008906073448, 9309805333008203501246, 987309241535765332024955809, 351345748109942610415182510895442, 459648902729700156671704473390158212154, 2067884865276847662816755891452805155809167114

Offset: 0

Author

Vaclav Kotesovec, Nov 19 2022

Keywords

• nonn

Crossrefs

Cf. A357871, A358496.

Programs

Mathematica

Table[Sum[Binomial[Binomial[n, k], n], {k, 0, n}], {n, 0, 16}]

PARI

a(n) = sum(k=0, n, binomial(binomial(n, k), n)); \\ Michel Marcus, Nov 19 2022

Formula

a(n) ~ c * binomial(binomial(n, n/2), n), where c = EllipticTheta[3,0,1/E^2] = JacobiTheta3(0,exp(-2)) = A218792 = 1.271341522189... if n is even and c = EllipticTheta[2,0,1/E^2] = JacobiTheta2(0,exp(-2)) = 1.23528676585389... if n is odd.

Equivalently, a(n) ~ c * 2^(n^2 + n/2 - 1/2) * exp(n - 1/4) / (Pi^((n+1)/2) * n^((3*n+1)/2)).