A155858 Diagonal sums of triangle A155856.

1, 1, 3, 9, 35, 168, 967, 6538, 50831, 446919, 4383861, 47451921, 561715093, 7217604520, 100031995789, 1487319385140, 23613262336093, 398673670050021, 7132188802005991, 134766129577134553, 2681929390235577831

Offset: 0

Author

Paul Barry, Jan 29 2009

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A155856.

Programs

Mathematica

Table[Sum[Binomial[2*n-3*k, k]*(n-2*k)!, {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Jun 05 2021 *)

Sage

[sum( binomial(2*n-3*k, k)*factorial(n-2*k) for k in (0..n//2) ) for n in (0..30)] # G. C. Greubel, Jun 05 2021

Formula

G.f.: 1/(1 -x^2 -x/(1 -x^2 -x/(1 -x^2 -2*x/(1 -x^2 -2*x/(1 -x^2 -3*x/(1 -x^2 -3*x/(1 - ... (continued fraction);

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k)*(n-2*k)!.

Conjecture: a(n) -(n-1)*a(n-1) -(n-2)*a(n-2) +(n-3)*a(n-3) +(n-10)*a(n-4) -5*a(n-5) +3*a(n-6) +3*a(n-7) = 0. - R. J. Mathar, Feb 05 2015

a(n) ~ n! * (1 + 2/n + 1/n^2 - 2/(3*n^3) - 22/(3*n^4) - 491/(15*n^5) - 11467/(90*n^6) - ...). - Vaclav Kotesovec, Jun 05 2021