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A217667 G.f.: Sum_{n>=0} (x + x^(2*n))^n. 3
1, 1, 2, 1, 1, 3, 1, 1, 5, 1, 1, 5, 1, 4, 6, 1, 1, 7, 8, 1, 8, 1, 1, 19, 1, 5, 10, 1, 16, 11, 1, 1, 23, 22, 1, 13, 1, 1, 42, 21, 1, 20, 1, 37, 16, 1, 36, 17, 46, 1, 34, 1, 1, 130, 1, 1, 20, 1, 67, 56, 85, 7, 22, 79, 1, 23, 1, 121, 185, 1, 1, 25, 23, 106, 191, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,

(2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),

for any fixed integer k; here, k = 1 and y = 1, F = x, G = x^2.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1000

FORMULA

Generating functions.

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

(2) Sum_{n>=0} x^(n^2) / (1 - x^(2*n+1))^(n+1). - Paul D. Hanna, Jun 02 2019

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 + x^7 + 5*x^8 + x^9 +...

where

A(x) = 1 + (x + x^2) + (x + x^4)^2 + (x + x^6)^3 + (x + x^8)^4 + (x + x^10)^5 +...

Also

A(x) = 1/(1-x) + x/(1 - x^3)^2 + x^4/(1 - x^5)^3 + x^9/(1 - x^7)^4 + x^16/(1 - x^9)^5 + x^25/(1 - x^11)^6 + x^36/(1 - x^13)^7 + x^49/(1 - x^15)^8 + ...

MATHEMATICA

terms = 100; Sum[(x + x^(2*n))^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-Fran├žois Alcover, May 16 2017 *)

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, (x+x^(2*m) +x*O(x^n))^m), n)}

for(n=0, 100, print1(a(n), ", "))

CROSSREFS

Cf. A217668, A217669.

Sequence in context: A301376 A307828 A280698 * A240850 A194086 A164659

Adjacent sequences:  A217664 A217665 A217666 * A217668 A217669 A217670

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 10 2012

STATUS

approved

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Last modified May 26 15:50 EDT 2020. Contains 334626 sequences. (Running on oeis4.)