A271235 G.f. equals the square root of P(4*x), where P(x) is the g.f. of the partition numbers (A000041).

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

Offset: 0

Author

Paul D. Hanna, Apr 02 2016

Keywords

• nonn

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

Examples

			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) +...
		

Links

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

Crossrefs

Cf. A298411, A298993, A298994, A370735.

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).

Programs

Mathematica

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

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

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