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A204275 G.f.: Product_{n>=1} (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203 is the companion Pell numbers. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A120767 A051771 A008318 * A160048 A019047 A221679

Adjacent sequences: A204272 A204273 A204274 * A204276 A204277 A204278

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 13 2012

STATUS

approved

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Last modified January 30 02:34 EST 2023. Contains 359939 sequences. (Running on oeis4.)