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A307496 Expansion of Product_{k>=1}  (1 + ((1 - sqrt(1 - 4*x))/2)^k). 1
1, 1, 2, 6, 18, 57, 187, 629, 2156, 7502, 26427, 94053, 337653, 1221260, 4445892, 16277089, 59893052, 221370725, 821499759, 3059620076, 11432831745, 42848889316, 161032785057, 606710026659, 2291156662259, 8670805904186, 32879697168622, 124910667052026, 475357627716839, 1811931609379926 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Catalan transform of A000009 (number of partitions into distinct parts).

LINKS

Table of n, a(n) for n=0..29.

FORMULA

G.f.: Product_{k>=1}  1/(1 - ((1 - sqrt(1 - 4*x))/2)^(2*k-1)).

Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000009.

a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*A000009(k) for n > 0.

MATHEMATICA

nmax = 29; CoefficientList[Series[Product[(1 + ((1 - Sqrt[1 - 4 x])/2)^k), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 29; CoefficientList[Series[Product[1/(1 - ((1 - Sqrt[1 - 4 x])/2)^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k PartitionsQ[k], {k, n}], {n, 29}]]

CROSSREFS

Cf. A000009, A100100, A286955.

Sequence in context: A126983 A104629 A000957 * A125305 A273203 A148458

Adjacent sequences:  A307493 A307494 A307495 * A307497 A307498 A307499

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 10 2019

STATUS

approved

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Last modified December 6 19:31 EST 2019. Contains 329809 sequences. (Running on oeis4.)