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A007696
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Quartic (or 4-fold) factorial numbers: a(n) = product(k=0..n-1, 4*k+1 ).
(Formerly M4001)
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47
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1, 1, 5, 45, 585, 9945, 208845, 5221125, 151412625, 4996616625, 184874815125, 7579867420125, 341094033905625, 16713607661375625, 885821206052908125, 50491808745015763125, 3080000333445961550625, 200200021673987500790625
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OFFSET
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0,3
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COMMENTS
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a(n), n>=1, enumerates increasing quintic (5-ary) trees. See a D. Callan comment on A007559 (number of increasing quarterny trees).
Hankel transform is A169619. [From Paul Barry, Dec 03 2009]
a(n) is the product of the positive integers k<=4*n that have k == 1 (modulo 4). - Peter Luschny, Jun 23 2011
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
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FORMULA
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E.g.f.: (1-4*x)^(-1/4).
a(n) ~ 2^(5/2)*pi^(1/2)*Gamma(1/4)^-1*n^(3/4)*2^(2*n)*e^-n*n^n*{1 + 23/96*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
a(n) = Sum_{k=0..n} (-4)^(n-k)*A048994(n, k) .- Philippe DELEHAM, Oct 29 2005
G.f.: 1/(1-x/(1-4x/(1-5x/(1-8x/(1-9x/(1-12x/(1-13x/(1-.../(1-A042948(n+1)*x/(1-... (continued fraction). [From Paul Barry, Dec 03 2009]
a(n)=(-3)^n*sum_{k=0..n} (4/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. [From Mircea Merca, MAy 03 2012]
G.f.: 1/T(0) where T(k) = 1 - x*(4*k+1)/(1 - x*(4*k+4)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: 1 + x/Q(0), where Q(k)= 1 + x + 2*(2*k-1)*x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k+1)/(x*(4*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013
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MAPLE
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x:='x'; G(x):=(1-4*x)^(-1/4): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1], x) od: seq(eval(f[n], x=0), n=0..17); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2009]
A007696 := n -> mul(k, k = select(k-> k mod 4 = 1, [$ 1 .. 4*n])): seq(A007696(n), n=0..17);
- Peter Luschny, Jun 23 2011
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MATHEMATICA
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s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 0, 5!, 4}]; lst [From Vladimir Joseph Stephan Orlovsky, Nov 08 2008]
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PROG
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(PARI) a(n) = prod(k=0, n-1, 4*k+1 );
(Maxima) A007696(n):=prod(4*k+1, k, 0, n-1)$
makelist(A007696(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
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CROSSREFS
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Cf. A001147, A007559, A034255, A004981, A047053, A001813, A051142. a(n)= A049029(n, 1), n >= 1 (first column of triangle).
Sequence in context: A121414 A097328 A051539 * A090136 A090356 A201365
Adjacent sequences: A007693 A007694 A007695 * A007697 A007698 A007699
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KEYWORD
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nonn,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Better description from Wolfdieter Lang, Dec 11 1999.
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STATUS
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approved
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