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A022571
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Expansion of Product_{m>=1} (1+x^m)^6.
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6
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1, 6, 21, 62, 162, 384, 855, 1806, 3648, 7110, 13434, 24702, 44361, 78006, 134592, 228302, 381300, 627840, 1020394, 1638528, 2601849, 4088780, 6363354, 9813504, 15005458, 22760262, 34261248, 51204222, 76005906, 112092438, 164296989, 239404860, 346898496, 499971968, 716906394
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OFFSET
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0,2
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REFERENCES
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A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", New York, Gordon and Breach Science Publishers, 1986-1992, p. 755, Eq. 6.2.2.2. MR0874986 (88f:00013)
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LINKS
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FORMULA
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Euler transform of period 2 sequence [6, 0, ...]. - Michael Somos, Jul 09 2005
Expansion of q^(-1/4)(eta(q^2)/eta(q))^6 in powers of q. - Michael Somos, Jul 09 2005
Expansion of q^(-1/4)(1/2)k^(1/2)/k' in powers of q. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(x)=(x*A(x^4))^4 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(4096uv+48u+1)v-u^2 . - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^2-v^2)^2 -uv(1+8uv)^2 . - Michael Somos, Jul 09 2005
G.f.: Product_{k>0} (1+x^k)^6.
a(n) ~ exp(Pi * sqrt(2*n)) / (16 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
G.f.: exp(6*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
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MATHEMATICA
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nmax=50; CoefficientList[Series[Product[(1+q^m)^6, {m, 1, nmax}], {q, 0, nmax}], q] (* Vaclav Kotesovec, Mar 05 2015 *)
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff( prod(k=1, n, 1+x^k, 1+x*O(x^n))^6, n)) /* Michael Somos, Jul 09 2005 */
(Magma) Coefficients(&*[(1+x^m)^6:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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