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%I #33 Feb 28 2024 18:09:55
%S 1,2,14,68,406,1820,10892,48008,266214,1248044,6454116,29642424,
%T 156638076,707729176,3551518936,16671232784,81685862790,375557689292,
%U 1843995831412,8437648295384,40779718859796,188104838512840,891508943457064,4091507664092016,19457793452994012,88760334081132280,415942096027738728,1905990594266105648,8875964207106121784,40416438507461834160
%N G.f. equals the square root of P(4*x), where P(x) is the g.f. of the partition numbers (A000041).
%C More formulas and information can be derived from entry A000041.
%C 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
%H Vaclav Kotesovec, <a href="/A271235/b271235.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: Product_{n>=1} 1/sqrt(1 - (4*x)^n).
%F 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.
%F a(n) ~ 4^(n-1) * exp(sqrt(n/3)*Pi) / (3^(3/8) * n^(7/8)). - _Vaclav Kotesovec_, Apr 02 2016
%e 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 +...
%e where A(x)^2 = P(4*x).
%e RELATED SERIES.
%e 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 +...
%e 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) +...
%t nmax = 30; CoefficientList[Series[Product[1/Sqrt[1 - (4*x)^k], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 02 2016 *)
%o (PARI) {a(n) = polcoeff( prod(k=1,n, 1/sqrt(1 - (4*x)^k +x*O(x^n))),n)}
%o for(n=0,30,print1(a(n),", "))
%o (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))};
%o for(n=0,30,print1(a(n),", "))
%o (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
%Y Cf. A298411, A298993, A298994, A370735.
%Y 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).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 02 2016
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