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A264925
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G.f.: 1 / Product_{n>=0} (1 - x^(n+5))^((n+1)*(n+2)*(n+3)*(n+4)/4!).
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4
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1, 0, 0, 0, 0, 1, 5, 15, 35, 70, 127, 215, 360, 605, 1080, 2003, 3890, 7570, 14715, 27960, 52255, 95705, 173295, 311060, 557400, 999032, 1795880, 3235130, 5835955, 10521060, 18931287, 33956485, 60692510, 108087835, 191883595, 339724144, 600203700, 1058605775, 1864535670, 3279862975, 5762287759, 10109925380
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OFFSET
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0,7
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COMMENTS
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Number of partitions of n objects of 5 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].
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LINKS
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FORMULA
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G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^5 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)*(d-3)*(d-4)/4!.
a(n) ~ Pi^(95/288) / (2 * 3^(527/576) * 7^(239/1728) * n^(1103/1728)) * exp(-25*Zeta'(-1)/12 - log(2*Pi)/2 + 595*Zeta(3)/(48*Pi^2) - 29291*Zeta(5) / (128*Pi^4) - 2480625 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) + 72930375 * Zeta(5)^3 / (2*Pi^14) - 1063324867500 * Zeta(5)^5/Pi^24 - 5*Zeta'(-3)/12 + (41 * 7^(1/6) * Pi/(768*sqrt(3)) - 2625 * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5)/(2*Pi^7) + 540225 * sqrt(3) * 7^(1/6) * Zeta(5)^2/(16*Pi^9) - 4740474375 * sqrt(3) * 7^(1/6) * Zeta(5)^4/(4*Pi^19)) * n^(1/6) + (-25 * 7^(1/3) * Zeta(3)/(4*Pi^2) + 735 * 7^(1/3) * Zeta(5) /(8*Pi^4) - 3969000 * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (7*sqrt(7/3)*Pi/24 - 4725 * sqrt(21) * Zeta(5)^2 / Pi^9) * sqrt(n) - 45 * 7^(2/3) * Zeta(5)/(2*Pi^4) * n^(2/3) + 2*sqrt(3)*Pi / (5*7^(1/6)) * n^(5/6)). - Vaclav Kotesovec, Dec 09 2015
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EXAMPLE
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G.f.: A(x) = 1 + x^5 + 5*x^6 + 15*x^7 + 35*x^8 + 70*x^9 + 127*x^10 + 215*x^11 + 360*x^12 +...
where
1/A(x) = (1-x^5) * (1-x^6)^5 * (1-x^7)^15 * (1-x^8)^35 * (1-x^9)^70 * (1-x^10)^126 * (1-x^11)^210 * (1-x^12)^330 * (1-x^13)^495 *...
Also,
log(A(x)) = (x/(1-x))^5 + (x^2/(1-x^2))^5/2 + (x^3/(1-x^3))^5/3 + (x^4/(1-x^4))^5/4 + (x^5/(1-x^5))^5/5 + (x^6/(1-x^6))^5/6 +...
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-4)*(k-3)*(k-2)*(k-1)/24), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)
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PROG
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(PARI) {a(n) = my(A=1); A = prod(k=0, n, 1/(1 - x^(k+4) +x*O(x^n) )^((k+1)*(k+2)*(k+3)/3!) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = exp( sum(k=1, n+1, (x^k/(1 - x^k))^4 /k +x*O(x^n) ) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
(PARI) {L(n) = sumdiv(n, d, d*(d-1)*(d-2)*(d-3)*(d-4)/4!)}
{a(n) = my(A=1); A = exp( sum(k=1, n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
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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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