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A209441 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(5*n) * Product_{k=1..n} (1 - 1/A(x)^k). 6
1, 1, 4, 30, 260, 2463, 24656, 256493, 2745149, 30031677, 334334789, 3775539592, 43145236171, 498018527632, 5798165437701, 68009060597311, 802908842472516, 9533509909631074, 113774810189434083, 1363985826416978416, 16418865502303963429, 198369001060550654651 (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)^n * Product_{k=1..n} (1 - 1/G(x)^k)

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

LINKS

Table of n, a(n) for n=0..21.

FORMULA

G.f. satisfies: 1+x = A(y) where y = x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15.

G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+11)/2) * Product_{k=1..n} (A(x)^k - 1).

EXAMPLE

G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 260*x^4 + 2463*x^5 + 24656*x^6 +...

The g.f. satisfies:

x = (A(x)-1)/A(x)^6 + (A(x)-1)*(A(x)^2-1)/A(x)^13 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)/A(x)^21 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)/A(x)^30 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)*(A(x)^5-1)/A(x)^40 +...

MATHEMATICA

nmax = 20; aa = ConstantArray[0, nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^m)/AGF^5, {m, 1, k}], {k, 1, j}], {x, 0, j}]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Dec 01 2014 *)

CoefficientList[1+InverseSeries[Series[x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 01 2014 *)

PROG

(PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15 +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)^(5*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]}

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

CROSSREFS

Cf. A001002, A181997, A181998, A209442, A214695 (variant).

Sequence in context: A091527 A201200 A102307 * A052658 A220442 A215698

Adjacent sequences:  A209438 A209439 A209440 * A209442 A209443 A209444

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 08 2012

STATUS

approved

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Last modified May 25 21:53 EDT 2017. Contains 287059 sequences.