A118393 -id:A118393 - cp's OEIS frontend

A118396 Eigenvector of triangle A118394; E.g.f.: exp( Sum_{n>=0} x^(3^n) ).

1, 1, 1, 7, 25, 61, 481, 2731, 10417, 454105, 4309921, 23452111, 592433161, 6789801877, 46254009985, 893881991731, 11548704851041, 93501748795441, 4828847934591937, 83867376656907415, 823025819684123641, 33409213329178701421, 640457721676922946721

Offset: 0

Paul D. Hanna, May 07 2006

nonn

E.g.f. of triangle A118394 is: exp(x+y*x^3), where A118394(n,k) = n!/k!/(n-3*k)!. More generally, given a triangle with e.g.f.: exp(x+y*x^b), the eigenvector will have e.g.f.: exp( Sum_{n>=0} x^(b^n) ).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A118394, A118395; variants: A118393, A118932.

a:= proc(n) option remember; `if`(n=0, 1, add((j-> j!*
      a(n-j)*binomial(n-1, j-1))(3^i), i=0..ilog[3](n)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 01 2017

a[n_] := a[n] = If[n==0, 1, Sum[Function[j, j! a[n-j] Binomial[n-1, j-1]][3^i], {i, 0, Log[3, n]}]];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

{a(n)=n!*polcoeff(exp(sum(k=0,ceil(log(n+1)/log(3)),x^(3^k))+x*O(x^n)),n)}

a(n) = Sum_{k=0..[n/3]} n!/k!/(n-3*k)! *a(k) for n>=0, with a(0)=1.

A374346 E.g.f. A(x) satisfies A(x) = A(x^2)^(1/2) * exp(2*x) with A(0)=1.

Original entry on oeis.org

1, 2, 6, 20, 88, 432, 2464, 14912, 111360, 912896, 8491264, 80905728, 861835264, 9524264960, 113218762752, 1362387243008, 20665650774016, 337892698226688, 6100999266304000, 106342541313572864, 2014622956858638336, 37864490015441027072

Offset: 0

Seiichi Manyama, Jul 05 2024

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A005388, A008683, A118393, A308392.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(2*sum(k=0, ceil(log(N+1)/log(2)), x^2^k/2^k))))

E.g.f.: exp( 2 * Sum_{k>=0} x^(2^k)/2^k ).

E.g.f.: 1/( Product_{k>=1} (1 - x^(2*k-1))^(mu(2*k-1)/(2*k-1)) )^2, where mu() is the Moebius function.

cp's OEIS Frontend

A118396 Eigenvector of triangle A118394; E.g.f.: exp( Sum_{n>=0} x^(3^n) ).

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