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 A277212 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5 in powers of x. 11
 1, 5, 20, 65, 190, 505, 1260, 2970, 6700, 14535, 30520, 62235, 123720, 240340, 457380, 854190, 1568230, 2834120, 5048140, 8871450, 15396690, 26410860, 44811440, 75254240, 125162100, 206275505, 337032360, 546183425, 878270360, 1401857550, 2221862260 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS In general, for fixed m > 1, if g.f. = Product_{k>=1} (1 - x^(m*k))/(1 - x^k)^m, then a(n, m) ~ exp(Pi*sqrt(2*n*(m-1/m)/3)) * (m^2 - 1)^(m/4) / (2^(3*m/4 + 1/2) * 3^(m/4) * m^(m/4 + 1/2) * n^(m/4 + 1/2)). - Vaclav Kotesovec, Nov 10 2016 LINKS Robert Israel, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, q-Pochhammer Symbol FORMULA G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5. a(n) ~ exp(4*Pi*sqrt(n/5)) / (sqrt(2) * 5^(7/4) * n^(7/4)). - Vaclav Kotesovec, Nov 10 2016 G.f.: (x^5; x^5)_inf/((x; x)_inf)^5, where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 20 2016 a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017 EXAMPLE G.f.: 1 + 5*x + 20*x^2 + 65*x^3 + 190*x^4 + 505*x^5 + 1260*x^6 + ... MAPLE N:= 100: # to get a(0)..a(N) S:= series(mul((1-x^(5*n))/(1-x^n)^5, n=1..N), x, N+1): seq(coeff(S, x, n), n=0..N); # Robert Israel, Nov 09 2016 MATHEMATICA nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *) (QPochhammer[x^5, x^5]/QPochhammer[x, x]^5 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *) PROG (PARI) first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(5*k))/(1-x^k)^5, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016 (PARI) x='x+O('x^66); Vec(eta(x^5)/eta(x)^5) \\ Joerg Arndt, Nov 27 2016 CROSSREFS Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), A274327 (k=4), this sequence (k=5), A160539 (k=7). Cf. A109064. Sequence in context: A100534 A285928 A160506 * A160528 A023004 A001873 Adjacent sequences:  A277209 A277210 A277211 * A277213 A277214 A277215 KEYWORD nonn AUTHOR Seiichi Manyama, Nov 07 2016 STATUS approved

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Last modified June 26 00:10 EDT 2019. Contains 324367 sequences. (Running on oeis4.)