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(1) log( Product_{n>=1} (1 - x^(2*n))/(1 - x^n)^3 ) = Sum_{n>=1} sigma(2*n) * x^n/n (see formula of _Joerg Arndt_ in A182818).

(2) log( C(x) ) = Sum_{n>=1} binomial(2*n,n)/2 * x^n/n, where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

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L.g.f: L(x) = 3*x + 21*x^2/2 + 120*x^3/3 + 525*x^4/4 + 2268*x^5/5 + 12936*x^6/6 + 41184*x^7/7 + 199485*x^8/8 + 948090*x^9/9 + 3879876*x^10/10 + 12697776*x^11/11 + ... + sigma(2*n) * binomial(2*n,n)/2 * x^n/n + ...

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Paul D. Hanna, <a href="/A322185/b322185.txt">Table of n, a(n) for n = 1..512</a>

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a(n) = sigma(2*n) * binomial(2*n,n)/2, for n >= 1.

3, 21, 120, 525, 2268, 12936, 41184, 199485, 948090, 3879876, 12697776, 81124680, 218412600, 1123264800, 5584230720, 18934032285, 63007367940, 412918656150, 1060357914000, 6203093796900, 25836377973120, 88372156476240, 296403506193600, 1999351428352200, 5878093199355468, 24300008114457096, 116816365538886720, 458921436045626400, 1353026992479346800

1,1

Related logarithmic series:

(1) log( Product_{n>=1} (1 - x^(2*n))/(1 - x^n)^3 ) = Sum_{n>=1} sigma(2*n) * x^n/n.

(2) log( C(x) ) = Sum_{n>=1} binomial(2*n,n)/2 * x^n/n, where C(x) = 1 + x*C(x)^2 is the Catalan function.

a(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - (x + y)^n) ), for n >= 1.

L.g.f: L(x) = 3*x + 21*x^2/2 + 120*x^3/3 + 525*x^4/4 + 2268*x^5/5 + 12936*x^6/6 + 41184*x^7/7 + 199485*x^8/8 + 948090*x^9/9 + 3879876*x^10/10 + 12697776*x^11/11 + ...

RELATED SERIES.

exp(L(x)) = 1 + 3*x + 15*x^2 + 76*x^3 + 357*x^4 + 1662*x^5 + 8203*x^6 + 36609*x^7 + 169800*x^8 + 788024*x^9 + 3586350*x^10 + 15948147*x^11 + ... + A322186(n)*x^n + ...

The table of coefficients of x^n*y^k/(n+k) in

log( Product_{n>=1} 1/(1 - (x + y)^n) ) = (1*x + 1*y)/1 + (3*x^2 + 6*x*y + 3*y^2)/2 + (4*x^3 + 12*x^2*y + 12*x*y^2 + 4*y^3)/3 + (7*x^4 + 28*x^3*y + 42*x^2*y^2 + 28*x*y^3 + 7*y^4)/4 + (6*x^5 + 30*x^4*y + 60*x^3*y^2 + 60*x^2*y^3 + 30*x*y^4 + 6*y^5)/5 + (12*x^6 + 72*x^5*y + 180*x^4*y^2 + 240*x^3*y^3 + 180*x^2*y^4 + 72*x*y^5 + 12*y^6)/6 + (8*x^7 + 56*x^6*y + 168*x^5*y^2 + 280*x^4*y^3 + 280*x^3*y^4 + 168*x^2*y^5 + 56*x*y^6 + 8*y^7)/7 + (15*x^8 + 120*x^7*y + 420*x^6*y^2 + 840*x^5*y^3 + 1050*x^4*y^4 + 840*x^3*y^5 + 420*x^2*y^6 + 120*x*y^7 + 15*y^8)/8 + ...

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n=0: [0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, ..., sigma(k), ...];

n=1: [1, 6, 12, 28, 30, 72, 56, 120, 117, 180, ...];

n=2: [3, 12, 42, 60, 180, 168, 420, 468, 810, 660, ...];

n=3: [4, 28, 60, 240, 280, 840, 1092, 2160, 1980, 6160, ...];

n=4: [7, 30, 180, 280, 1050, 1638, 3780, 3960, 13860, 10010, ...];

n=5: [6, 72, 168, 840, 1638, 4536, 5544, 22176, 18018, 48048, ...];

n=6: [12, 56, 420, 1092, 3780, 5544, 25872, 24024, 72072, 120120, ...];

n=7: [8, 120, 468, 2160, 3960, 22176, 24024, 82368, 154440, 354640, ...];

n=8: [15, 117, 810, 1980, 13860, 18018, 72072, 154440, 398970, 437580, ...];

n=9: [13, 180, 660, 6160, 10010, 48048, 120120, 354640, 437580, 1896180, ...];

n=10: [18, 132, 1848, 4004, 24024, 72072, 248248, 350064, 1706562, 1847560, ...]; ...

in which the diagonal of coefficients of x^n*y^n/(2*n) equals

[0, 6, 42, 240, 1050, 4536, 25872, 82368, 398970, 1896180, ..., 2*a(n), ...],

which is twice this sequence.

(PARI) {a(n) = sigma(2*n) * binomial(2*n, n)/2}

for(n=1, 30, print1( a(n), ", ") )

(PARI) /* [x^n*y^n/n] log( Product_{n>=1} 1/(1 - (x + y)^n) ) */

N=30

{L = sum(n=1, 2*N+1, -log(1 - (x + y)^n +x*O(x^(2*N)) +y*O(y^(2*N))) ); }

{a(n) = polcoeff( n*polcoeff( L, n, x), n, y)}

for(n=1, N, print1( a(n), ", ") )

Cf. A322186, A322203, A322187.

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Paul D. Hanna, Dec 07 2018

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