A318248 -id:A318248 - cp's OEIS frontend

A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^k).

1, 1, 2, 4, 8, 15, 31, 57, 113, 212, 410, 757, 1464, 2684, 5083, 9380, 17569, 32120, 59977, 109193, 202046, 367951, 675541, 1224453, 2243795, 4052369, 7377243, 13314989, 24140198, 43406515, 78510429, 140800279, 253663615, 454352111, 815790813, 1457485309

Offset: 0

Vaclav Kotesovec, Mar 08 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500

Cf. A000045, A023882, A075900, A077365, A179381, A318248.

nmax = 40; CoefficientList[Series[Product[1/(1-Fibonacci[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

log(a(n)) ~ log(phi)*n + 2*sqrt(polylog(2, 1/sqrt(5))*n) - 3*(log(n)/4), where polylog(2, 1/sqrt(5)) = 0.5107013915606224266804289751265205446721... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio.

A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108.

Original entry on oeis.org

1, 1, 2, 7, 19, 66, 212, 743, 2487, 9012, 31177, 113775, 404584, 1490726, 5376676, 20028981, 73068861, 273659672, 1009921813, 3801386137, 14125670266, 53477758556, 199950414035, 759566205693, 2857261603610, 10889590477287, 41136917417501, 157329747348492

Offset: 0

Vaclav Kotesovec, Aug 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1650

Cf. A000108, A022629, A179381, A318248, A318263.

C:= proc(n) option remember; binomial(n+n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2019

nmax = 40; CoefficientList[Series[Product[1+CatalanNumber[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += CatalanNumber[k]*poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

a(n) ~ c * A000108(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + C(k)/4^k) = 2.608465265690846547082817204714986077801494... - Vaclav Kotesovec, Aug 24 2018

A318263 Expansion of Product_{k>=1} (1 + Lucas(k)*x^k).

Original entry on oeis.org

1, 1, 3, 7, 11, 30, 62, 129, 235, 541, 1034, 2101, 4140, 8129, 15984, 31903, 60398, 117646, 228808, 433768, 836552, 1601282, 3031299, 5736396, 10899112, 20466182, 38556342, 72522116, 135662847, 253047629, 473785878, 878655661, 1634304062, 3033385668, 5608183925

Offset: 0

Vaclav Kotesovec, Aug 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500

Cf. A000032, A022629, A318248, A318264.

nmax = 40; CoefficientList[Series[Product[1+LucasL[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += LucasL[k]*poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

From Vaclav Kotesovec, Aug 24 2018: (Start)
a(n) ~ c * A000032(n) * A000009(n) ~ c * phi^n * exp(Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)), where c = Product_{k>=1} ((1 + Lucas(k)/phi^k)/2) = 0.8503149035690839100210269103058319341315494385103929947491... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
Equivalently, c = Product_{k>=1} (1 + (-1)^k/(2*phi^(2*k))),
c = 2/3 * QPochhammer[-1/2, -1/GoldenRatio^2]. (End)

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A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^k).

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A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108.

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A318263 Expansion of Product_{k>=1} (1 + Lucas(k)*x^k).

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