A122920 Diagonal sums of number triangle A122919.

1, 1, 4, 12, 39, 129, 436, 1498, 5218, 18386, 65420, 234734, 848403, 3086001, 11288412, 41499354, 153247278, 568188606, 2114334312, 7893906144, 29561195238, 111007927386, 417918303144, 1577061975492, 5964172347604, 22601012748124, 85806694043116, 326343785428946, 1243200250005995

Offset: 0

Author

Paul Barry, Sep 19 2006

Keywords

• easy
• nonn

Comments

Starting with offset 1 = iterates of M * [1,1,1,0,0,0,...] where M is the tridiagonal matrix with [0,2,2,2,...] as the main diagonal and [1,1,1,...] as the super and subdiagonals. - Gary W. Adamson, Jan 09 2009

Partial sums are Fine numbers (A000957) with offset 3. - Alexander Burstein, Apr 15 2015

Examples

			G.f. = 1 + x + 4*x^2 + 12*x^3 + 39*x^4 + 129*x^5 + 436*x^6 + 1498*x^7 + 5218*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

Crossrefs

Cf. A000957.

Programs

Mathematica

CoefficientList[Series[((1-x)*(1-2*x-2*x^2-Sqrt[1-4*x])/(2*(2+x)*x^3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)
Table[9/(16 (-2)^n) + 3 (2n+4)! HypergeometricPFQ[{1, n+5/2, n+3}, {n+2, n+5}, -8]/((n+1)! (n+4)!), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 26 2015 *)

PARI

x='x+O('x^66); Vec(((1-x)*(1-2*x-2*x^2-sqrt(1-4*x))/(2*(2+x)*x^3))) \\ Joerg Arndt, May 08 2013

Formula

G.f.: ((1-x)*(1-2*x-2*x^2-sqrt(1-4*x))/(2*(2+x)*x^3)).

Conjecture: 2*n*(n+3)*a(n) - (7*n^2+9*n+4)*a(n-1) - 2*(n+1)*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Nov 05 2012

a(n) ~ 2^(2*n+4) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 03 2014

From Vladimir Reshetnikov, Oct 26 2015: (Start)

a(n) = 9/(16*(-2)^n) + 3*(2*n+4)!*hypergeom([1,n+5/2,n+3], [n+2,n+5], -8)/((n+1)!*(n+4)!).

a(n) = 9/(16*(-2)^n) + 8*2^n*(2*n+5)!!*hypergeom([1,n+7/2], [n+5], -8)/(n+4)! - 4*2^n*(2*n+3)!!*hypergeom([1,n+5/2], [n+4], -8)/(n+3)!. (End)

G.f. A(x) =: y satisfies 0 = (1 - x)^2 - y*(1 - 3*x + 2*x^3) + y^2*(2*x^3 + x^4). - Michael Somos, Oct 26 2015

0 = a(n)*(+16*a(n+1) - 26*a(n+2) - 98*a(n+3) + 36*a(n+4)) + a(n+1)*(+50*a(n+1) + 35*a(n+2) - 179*a(n+3) + 46*a(n+4)) + a(n+2)*(+105*a(n+2) + 47*a(n+3) - 50*a(n+4)) + a(n+3)*(+14*a(n+3) + 4*a(n+4)) for all n>=0. - Michael Somos, Oct 26 2015