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A271235 G.f. equals the square root of P(4*x), where P(x) is the g.f. of the partition numbers (A000041). 6
1, 2, 14, 68, 406, 1820, 10892, 48008, 266214, 1248044, 6454116, 29642424, 156638076, 707729176, 3551518936, 16671232784, 81685862790, 375557689292, 1843995831412, 8437648295384, 40779718859796, 188104838512840, 891508943457064, 4091507664092016, 19457793452994012, 88760334081132280, 415942096027738728, 1905990594266105648, 8875964207106121784, 40416438507461834160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More formulas and information can be derived from entry A000041.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

FORMULA

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

Sum_{k=0..n} a(k) * a(n-k)  =  4^n * A000041(n), for n>=0, where A000041(n) equals the number of partitions of n.

a(n) ~ 4^(n-1) * exp(sqrt(n/3)*Pi) / (3^(3/8) * n^(7/8)). - Vaclav Kotesovec, Apr 02 2016

EXAMPLE

G.f.: A(x) = 1 + 2*x + 14*x^2 + 68*x^3 + 406*x^4 + 1820*x^5 + 10892*x^6 + 48008*x^7 + 266214*x^8 + 1248044*x^9 + 6454116*x^10 +...

where A(x)^2 = P(4*x).

RELATED SERIES.

P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + 101*x^13 + 135*x^14 +...+ A000041(n)*x^n +...

1/A(x)^6 = 1 - 12*x + 320*x^3 - 28672*x^6 + 9437184*x^10 - 11811160064*x^15  + 57174604644352*x^21 +...+ (-1)^n*(2*n+1)*(4*x)^(n*(n+1)/2) +...

MATHEMATICA

nmax = 30; CoefficientList[Series[Product[1/Sqrt[1 - (4*x)^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 02 2016 *)

PROG

(PARI) {a(n) = polcoeff( prod(k=1, n, 1/sqrt(1 - (4*x)^k +x*O(x^n))), n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n+1), (4*x)^(k^2) / prod(j=1, k, 1 - (4*x)^j, 1 + x*O(x^n))^2, 1)^(1/2), n))};

for(n=0, 30, print1(a(n), ", "))

(PARI) N=99; x='x+O('x^N); Vec(prod(k=1, N, 1/(1-(4*x)^k)^(1/2))) \\ Altug Alkan, Apr 20 2018

CROSSREFS

Cf. A298411, A298993, A298994.

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), this sequence (b=2), A271236 (b=3), A303135 (b=4), A303136 (b=5).

Sequence in context: A197608 A325925 A084132 * A084770 A086243 A258138

Adjacent sequences:  A271232 A271233 A271234 * A271236 A271237 A271238

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 02 2016

STATUS

approved

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Last modified September 22 03:21 EDT 2020. Contains 337289 sequences. (Running on oeis4.)