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Search: id:A064613
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| A064613 |
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Generalized binomial transform of Catalan numbers. |
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+0
7
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| 1, 3, 10, 37, 150, 654, 3012, 14445, 71398, 361114, 1859628, 9716194, 51373180, 274352316, 1477635912, 8016865533, 43773564294, 240356635170, 1326359740956, 7351846397334, 40913414754324, 228508350629892
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Second binomial transform of Catalan numbers. Exponential convolution of Catalan numbers and powers of 2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 03 2004
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
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FORMULA
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In Maple notation: a(n)=sum(binomial(n, k)*binomial(2*k, k)*2^(n-k)/(k+1), k = 0 .. n) = 2^n*hypergeom([1/2, -n], [2], -2), n=0, 1, ...
G.f.: (1-sqrt((1-6*x)/(1-2*x)))/2/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 03 2003
With offset 1 : a(1)=1, a(n)=2^(n-1)+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2004
(n+1)*a(n) = (8*n-2)*a(n-1)-(12*n-12)*a(n-2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 16 2004
E.g.f.: exp(4*x)*(BesselI(0, 2*x)-BesselI(1, 2*x)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 03 2004
Inverse binomial transform of A104455 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 30 2007
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CROSSREFS
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Cf. A007317, A000108, A014318.
Sequence in context: A063029 A044048 A086444 this_sequence A138378 A005493 A123636
Adjacent sequences: A064610 A064611 A064612 this_sequence A064614 A064615 A064616
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KEYWORD
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nonn
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AUTHOR
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Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 24 2001
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