A053297 Row sums of array T in A053199.

1, 5, 22, 92, 372, 1468, 5688, 21728, 82064, 307088, 1140320, 4206912, 15434048, 56350912, 204875648, 742104064, 2679197952, 9644109056, 34623075840, 124001176576, 443136848896, 1580464036864, 5626501838848, 19996918849536, 70960191213568, 251445325991936

Offset: 1

Author

Clark Kimberling, Mar 18 2000

Keywords

• nonn

Comments

The generating series is a power series composition G(F(t)) where F(t) = t + 3*t^2 + 7*t^3 + 15*t^4 + ... is generating series of A000225, and G(t) = t + 2*t^2 + 3*t^3 + 4*t^4 + ... is generating series of the natural numbers A000027. Proof follows as in reference below. - Oboifeng Dira, Nov 03 2016

Examples

			G.f. = x + 5*x^2 + 22*x^3 + 92*x^4 + 372*x^5 + 1468*x^6 + 5688*x^7 + 21728*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• O. Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics . 2017, Vol. 41 Issue 6, pp. 849-853.

Crossrefs

Cf. A000027, A000225, A053199.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2)); // G. C. Greubel, May 24 2018

Mathematica

Drop[CoefficientList[Series[x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2, {x,0,50}], x], 1] (* G. C. Greubel, May 24 2018 *)

PARI

my(x='x+O('x^30)); Vec(x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2) \\ G. C. Greubel, May 24 2018

Formula

G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x + 2*x^2)^2. - Michael Somos, Nov 03 2016

a(n) = 8*a(n-1) + 20*a(n-2) - 16*a(n-3) + 4*a(n-4) for all n in Z. - Michael Somos, Nov 03 2016

a(n) = -a(-n) * 2^n for all n in Z. - Michael Somos, Nov 03 2016