A139377 -id:A139377 - cp's OEIS frontend

A135582 Expansion of 1/((1-x^2*c(x))(1-x-2x^2)) where c(x) is the g.f. of A000108.

1, 1, 4, 7, 18, 39, 95, 232, 606, 1663, 4839, 14807, 47330, 156611, 532308, 1846622, 6507103, 23210020, 83590477, 303425693, 1108650850, 4073443378, 15039391464, 55763147423, 207543422052, 775082175863, 2903508757053, 10907257755616

Offset: 0

Paul Barry, Apr 15 2008

Diagonal sums of the Jacobsthal-Catalan triangle A139377.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000108, A001045, A132364.

Jacobsthal[n_]:= (2^n - (-1)^n)/3; g[0]:= 1; g[n_] := Sum[(i/(n - i)) * Binomial[2*n - 3*i - 1, n - 2*i], {i, 0, Floor[n/2]}]; a[n_]:= Sum[Jacobsthal[k + 1]*g[n - k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* G. C. Greubel, Oct 20 2016 *)

a(n) = Sum_{k=0..n} J(k+1)*A132364(n-k) where J(n)=A001045(n), Jacobsthal numbers.

Conjecture: (-n+1)*a(n) +6*(n-2)*a(n-1) +(-7*n+19)*a(n-2) +(-7*n+13)*a(n-3) +(13*n-31)*a(n-4) +2*(-n+4)*a(n-5) +4*(-2*n+5)*a(n-6)=0. - R. J. Mathar, Feb 23 2015

A139379 A Jacobsthal Catalan convolution.

Original entry on oeis.org

0, 1, 2, 6, 15, 41, 113, 327, 982, 3066, 9892, 32820, 111390, 385042, 1350722, 4795246, 17191535, 62139697, 226167557, 828085651, 3047683955, 11267975677, 41829610607, 155848125601, 582566960465, 2184167358991, 8211247681373

Offset: 0

Paul Barry, Apr 15 2008

Second column of A139377. Row sums of A139377 give a(n+1).

G.f.: xc(x)/(1-x-2x^2), c(x) the g.f. of A000108; a(n)=sum{k=0..n, A000108(k)*A001045(n-k)};

D-finite with recurrence n*a(n) +(6-5*n)*a(n-1) +2*(n-3)*a(n-2) +4*(2*n-3)*a(n-3)=0. - R. J. Mathar, Oct 25 2012

cp's OEIS Frontend

A135582 Expansion of 1/((1-x^2*c(x))(1-x-2x^2)) where c(x) is the g.f. of A000108.

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