A228365 Inverse binomial transform of the Galois numbers G_(n)^{(3)} (A006117).

1, 1, 3, 15, 129, 1833, 43347, 1705623, 112931553, 12639552945, 2413134909507, 788041911546303, 442817851480763169, 428369525248261655193, 716160018275094098267859, 2067365673240491189928496263, 10333740296321620864171488891201

Offset

0,3

Comments

Analog of the inverse binomial transform of G_(n)^{(q)} with q=2, A135922.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..91

Formula

a(n) ~ c * 3^(n^2/4), where c = EllipticTheta[3,0,1/3]/QPochhammer[1/3,1/3] = 3.019783845699... if n is even and c = EllipticTheta[2,0,1/3]/QPochhammer[1/3,1/3] = 3.01826904637117... if n is odd. - Vaclav Kotesovec, Aug 23 2013

Maple

b:= proc(n) option remember; add(mul(
      (3^(i+k)-1)/(3^i-1), i=1..n-k), k=0..n)
    end:
a:= proc(n) option remember;
      add(b(n-j)*binomial(n, j)*(-1)^j, j=0..n)
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Sep 24 2019

Mathematica

Table[SeriesCoefficient[Sum[x^n/Product[(1-(3^k-1)*x), {k, 0, n}], {n, 0, nn}], {x, 0, nn}] , {nn, 0, 20}] (* Vaclav Kotesovec, Aug 23 2013 *)

Crossrefs

Cf. A135922, A006117, A006116.

Sequence in context: A335390 A075475 A074241 * A117694 A264795 A222390

Adjacent sequences: A228362 A228363 A228364 * A228366 A228367 A228368

Keyword

nonn

Author

R. J. Mathar, Aug 21 2013

Status

approved