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Paul D. Hanna, <a href="/A355352/b355352.txt">Table of n, a(n) for n = 0..400</a>

allocated for Paul D. Hanna

G.f. A(x) satisfies: 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 2, 10, 50, 248, 1294, 7092, 40426, 236698, 1412860, 8561906, 52546920, 326011118, 2041512624, 12886608654, 81908498582, 523780469070, 3367399778356, 21752611767804, 141118852010146, 919035717462824, 6006146649948722, 39376700396145616, 258907024677687808

0,2

a(n) = Sum_{k=0..n} A355350(n,k) * 2^k for n >= 0.

G.f. A(x) satisfies:

(1) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

(2) 2*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

G.f.: A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 248*x^4 + 1294*x^5 + 7092*x^6 + 40426*x^7 + 236698*x^8 + 1412860*x^9 + 8561906*x^10 + ...

where

2*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...

also,

2*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...

where P(x) is the partition function and begins

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 + ... + A000041(n)*x^n + ...

(PARI) {a(n) = my(A=[1, 2], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));

A[#A] = -polcoeff( sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}

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

Cf. A355350, A355351, A355353, A355354, A355355, A355356, A355357.

allocated

nonn

Paul D. Hanna, Jun 29 2022

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editing

allocated for Paul D. Hanna

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