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A214693 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)). 8
1, 1, 4, 34, 338, 3691, 42623, 510949, 6289912, 78972928, 1006665781, 12985611054, 169115724583, 2219614920740, 29318819296959, 389331204757856, 5192978617937181, 69522908878900079, 933674035184058960, 12571898958515379108, 169651868248129552194 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare the g.f. to the identity:

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

which holds for all power series G(x) such that G(0)=1.

LINKS

Table of n, a(n) for n=0..20.

FORMULA

G.f. satisfies: 1+x = A(y) where y = x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9, which is the g.f. of row 3 in triangle A214690.

G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+6)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).

EXAMPLE

G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 338*x^4 + 3691*x^5 + 42623*x^6 +...

The g.f. satisfies:

x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^3-1)/A(x)^16 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^27 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^40 +

(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^55 +...

PROG

(PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9 +x^2*O(x^n)), n))}

(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(6*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A214690, A214692, A214694, A214695, A181997 (variant).

Sequence in context: A025572 A093137 A332617 * A107350 A206180 A274344

Adjacent sequences:  A214690 A214691 A214692 * A214694 A214695 A214696

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 26 2012

STATUS

approved

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Last modified February 18 21:20 EST 2020. Contains 332028 sequences. (Running on oeis4.)