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A047056
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Quintuple factorial numbers: Product_{k=0..n-1} (5*k+3).
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15
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1, 3, 24, 312, 5616, 129168, 3616704, 119351232, 4535346816, 195019913088, 9360955828224, 496130658895872, 28775578215960576, 1812861427605516288, 123274577077175107584, 8999044126633782853632, 701925441877435062583296, 58259811675827110194413568
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: (1-5*x)^(-3/5).
a(n) ~ sqrt(2*Pi)/Gamma(3/5)*n^(1/10)*(5*n/e)^n*(1 - 11/300*n^ - 1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
G.f.: 1/(1-3*x/(1-5*x/(1-8*x/(1-10*x/(1-13*x/(1-15*x/(1-18*x/(1-20*x/(1- ... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-2)^n*Sum_{k=0..n} (5/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/G(0) where G(k) = 1 - x*(5*k+3)/( 1 - 5*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - (5*k+3)*x/((5*k+3)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
D-finite with recurrence: a(n) +(-5*n+2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/5^2)^(1/5)*(Gamma(3/5) - Gamma(3/5, 1/5)). - Amiram Eldar, Dec 19 2022
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MAPLE
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seq(mul(5*k+3, k = 0..n-1), n = 0..20); # G. C. Greubel, Aug 20 2019
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MATHEMATICA
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Table[5^n*Pochhammer[3/5, n], {n, 0, 20}] (* G. C. Greubel, Aug 20 2019 *)
Join[{1}, FoldList[Times, 5*Range[0, 20]+3]] (* Harvey P. Dale, Oct 08 2020 *)
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PROG
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(PARI) vector(20, n, n--; prod(j=0, n-1, 5*j+3) ) \\ G. C. Greubel, Aug 20 2019
(Magma) [1] cat [(&*[5*k+3: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 20 2019
(Sage) [5^n*rising_factorial(3/5, n) for n in (0..20)] # G. C. Greubel, Aug 20 2019
(GAP) List([0..20], n-> Product([0..n-1], k-> 5*k+3 )); # G. C. Greubel, Aug 20 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Joe Keane (jgk(AT)jgk.org)
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STATUS
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approved
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