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A112700
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Partial sum of Catalan numbers A000108 multiplied by powers of 6.
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1
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1, 7, 79, 1159, 19303, 345895, 6504487, 126597031, 2528447911, 51526205863, 1067116097959, 22394503831975, 475191351108007, 10177980935594407, 219758235960500647, 4778128782752211367, 104526001924311998887
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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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a(n)=sum(C(k)*6^k, k=0..n), n>=0, with C(n):=A000108(n).
G.f.: c(6*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
Conjecture: (n+1)*a(n) +(-25*n+11)*a(n-1) +12*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 08 2016, verified by Robert Israel, Jun 28 2018
0 = a(n)*(+576*a(n+1) -636*a(n+2) +60*a(n+3)) +a(n+1)*(-564*a(n+1) +613*a(n+2) -61*a(n+3)) +a(n+2)*(+11*a(n+2) +a(n+3)) for all n>=0. - Michael Somos, Jun 28 2018
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MAPLE
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ListTools:-PartialSums([seq(binomial(2*n, n)/(n+1)*6^n, n=0..50)]); # Robert Israel, Jun 28 2018
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CROSSREFS
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Seventh column (m=6) of triangle A112705.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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