A186239 G.f. A(x) satisfies A(x) = 1+x*A(x)+x^2*A(x)^2+2*x^3*A(x)^3.

1, 1, 2, 6, 17, 51, 163, 533, 1779, 6055, 20908, 73052, 257863, 918139, 3293538, 11891778, 43183835, 157616827, 577902846, 2127539802, 7861397403, 29145629141, 108385383175, 404184619545, 1511132059333, 5663069529201, 21269203639596, 80044555061812

Offset

0,3

Links

Table of n, a(n) for n=0..27.

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

a(n) = 1/n * sum(j=0..n, C(n,j) * sum(i=j..n+j-1, C(j,i-j) * C(n-j,3*j-n-i-1) * 2^(3*j-n-i-1))), n>0.

Conjecture: 4*(2*n+3)*(n+1)*a(n) +(113*n^2-91*n-72)*a(n-1) + 3*(-135*n^2+263*n-108)*a(n-2) -3*(107*n-119)*(n-2)*a(n-3) -1411*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 14 2011

a(n) is the top left term of M^n, M = an infinite matrix with (1,1,1,...) as diagonals starting at positions (1,2), (1,1), and (2,1); with a diagonal of (2,2,2,...) starting at (3,1). - Gary W. Adamson, Nov 25 2011

Example

a(3) = 6 since the top row of M^3 = (6, 5, 3, 1, 0, 0, ...).

Mathematica

terms = 28;
A[_] = 0;
Do[A[x_] = 1 + x A[x] + x^2 A[x]^2 + 2 x^3 A[x]^3 + O[x]^terms, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 08 2018 *)

Crossrefs

Sequence in context: A157002 A071717 A181665 * A148451 A148452 A307975

Adjacent sequences: A186236 A186237 A186238 * A186240 A186241 A186242

Keyword

nonn

Author

Vladimir Kruchinin, Feb 15 2011

Status

approved