A204275 - OEIS

A204275

G.f.: Product_{n>=1} (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203 is the companion Pell numbers.

1, 2, 5, 26, 57, 222, 698, 2096, 6038, 19730, 58915, 169952, 516024, 1484958, 4397513, 13029558, 37094682, 106442928, 311875984, 879620854, 2522107990, 7229956352, 20398904648, 57543374566, 163053304047, 457604617760, 1283583473614, 3606627675050

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OFFSET

0,2

COMMENTS

Analog to Euler's identity: Product_{n>=1} (1+x^n) = Product_{n>=1} 1/(1-x^(2*n-1)), which is the g.f. for the number of partitions of distinct parts.

LINKS

Table of n, a(n) for n=0..27.

Eric Weisstein's World of Mathematics, Euler Identity.

FORMULA

G.f.: Product_{n>=1} 1/(1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)).

G.f.: exp( Sum_{n>=1} A000593(n) * A002203(n) * x^n/n ) where A000593(n) = sum of odd divisors of n.

a(n) = (1/n)*Sum_{k=1..n} A000593(k) * A002203(k)*a(n-k) for n>0, with a(0) = 1.

EXAMPLE

G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 57*x^4 + 222*x^5 + 698*x^6 +...

where A(x) = (1+2*x-x^2) * (1+6*x^2+x^4) * (1+14*x^3-x^6) * (1+34*x^4+x^8) * (1+82*x^5-x^10) * (1+198*x^6+x^12) *...* (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) *...

and 1/A(x) = (1-2*x-x^2) * (1-14*x^3-x^6) * (1-82*x^5-x^10) * (1-478*x^7-x^14) * (1-2786*x^9-x^18) * (1-16238*x^11-x^22) *...* (1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)) *...

Also, the logarithm of the g.f. equals the series:

log(A(x)) = 1*2*x + 1*6*x^2/2 + 4*14*x^3/3 + 1*34*x^4/4 + 6*82*x^5/5 + 4*198*x^6/6 + 8*478*x^7/7 + 1*1154*x^8/8 +...+ A000593(n)*A002203(n)*x^n/n +...

The companion Pell numbers (starting at offset 1) begin:

A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...]

and form the logarithm of a g.f. for Pell numbers:

log(1/(1-2*x-x^2)) = 2*x + 6*x^2/2 + 14*x^3/3 + 34*x^4/4 + 82*x^5/5 +...

PROG

(PARI) /* Subroutine used in PARI programs below: */

{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

(PARI) {a(n)=polcoeff(prod(k=1, n, 1+A002203(k)*x^k+(-1)^k*x^(2*k) +x*O(x^n)), n)}

(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-A002203(2*k-1)*x^(2*k-1)-x^(4*k-2) +x*O(x^n)), n)}

(PARI) /* Exponential form using sum of odd divisors of n: */

{A000593(n)=if(n<1, 0, sumdiv(n, d, (-1)^(d+1)*n/d))}

{a(n)=polcoeff(exp(sum(k=1, n, A000593(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

CROSSREFS

Cf. A203801, A204270, A000129 (Pell), A002203 (companion Pell), A000593.

Sequence in context: A051771 A008318 A365324 * A160048 A019047 A221679

Adjacent sequences: A204272 A204273 A204274 * A204276 A204277 A204278

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 13 2012

STATUS

approved