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A302616 G.f.: Sum_{n>=0} (1+x)^n * ((1+x)^n + (1+2*x)^n)^n / (2*(1+x)^n + (1+2*x)^n)^(n+1). 1
1, 1, 5, 47, 641, 11283, 243755, 6236425, 184344339, 6180934293, 231761841775, 9609095960569, 436486983640191, 21556547150620421, 1149991421265821805, 65903887072762826847, 4037804462230246970067, 263376035279468797850997, 18222095466457124888031163, 1332861882984996470788507485, 102768790354267787018489100069, 8330655428164879820409112566087 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

G.f. A(x) = Sum_{n>=0} a(n)*x^n equals:

(1) Sum_{n>=0} (1+x)^n * ((1+2*x)^n + (1+x)^n)^n / (2*(1+x)^n + (1+2*x)^n)^(n+1).

(2) Sum_{n>=0} (1+x)^n * ((1+2*x)^n - (1+x)^n)^n / (2*(1+x)^n - (1+2*x)^n)^(n+1).

EXAMPLE

G.f.: A(x) = 1 + x + 5*x^2 + 47*x^3 + 641*x^4 + 11283*x^5 + 243755*x^6 + 6236425*x^7 + 184344339*x^8 + 6180934293*x^9 + ...

such that

A(x) = 1/3  +  (1+x)*((1+x) + (1+2*x))/(2*(1+x) + (1+2*x))^2  +  (1+x)^2*((1+x)^2 + (1+2*x)^2)^2/(2*(1+x)^2 + (1+2*x)^2)^3  +  (1+x)^3*((1+x)^3 + (1+2*x)^3)^3/(2*(1+x)^3 + (1+2*x)^3)^4  +  (1+x)^4*((1+x)^4 + (1+2*x)^4)^4/(2*(1+x)^4 + (1+2*x)^4)^5  + ...

Also,

A(x) = 1  +  (1+x)*((1+2*x) - (1+x))/(2*(1+x) - (1+2*x))^2  +  (1+x)^2*((1+2*x)^2 - (1+x)^2)^2/(2*(1+x)^2 - (1+2*x)^2)^3  +  (1+x)^3*((1+2*x)^3 - (1+x)^3)^3/(2*(1+x)^3 - (1+2*x)^3)^4  +  (1+x)^4*((1+2*x)^4 - (1+x)^4)^4/(2*(1+x)^4 - (1+2*x)^4)^5  + ...

PROG

(PARI) {a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, (1+x)^m*((1+2*x)^m - (1+x)^m +o)^m/(2*(1+x)^m - (1+2*x)^m +o)^(m+1)); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Sequence in context: A328032 A074192 A058806 * A006902 A180254 A127696

Adjacent sequences:  A302613 A302614 A302615 * A302617 A302618 A302619

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 11 2018

STATUS

approved

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Last modified October 25 11:29 EDT 2021. Contains 348251 sequences. (Running on oeis4.)