A307496 Expansion of Product_{k>=1} (1 + ((1 - sqrt(1 - 4*x))/2)^k).

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

Offset: 0

Author

Ilya Gutkovskiy, Apr 10 2019

Keywords

• nonn

Comments

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

From Peter Bala, Jan 27 2020: (Start)

This sequence is the main diagonal of the lower triangular array formed by putting A000009 in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.

1

1 1

1 2 2

2 4 6 6

2 6 12 18 18

3 9 21 39 57 57

...

Alternatively, the sequence can be obtained by multiplying A000009 by the array A106566.

(End)

Crossrefs

Cf. A000009, A100100, A286955, A106566, A307495.

Programs

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}]]

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.

a(n) ~ c * 4^n / n^(3/2), where c = 1/sqrt(Pi) * Sum_{k>=0} k*A000009(k)/2^(k+1) = 1.12333545392999500455446757207126193339498222754079045166328600452997969... - Vaclav Kotesovec, Jan 28 2020, extended Aug 01 2022