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Search: id:A100192
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| A100192 |
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Sum binomial(2n,n+k)2^k, k=0..n. |
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+0
4
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| 1, 4, 18, 82, 374, 1704, 7752, 35214, 159750, 723880, 3276908, 14821668, 66991436, 302605528, 1366182276, 6165204102, 27811282374, 125415953208, 565408947756, 2548400193852, 11483706241044, 51739037228688, 233070330199296
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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A transform of 2^n under the mapping g(x)->(1/sqrt(1-4x))g(xc(x)^2), where c(x) is the g.f. of the Catalan numbers A000108. A transform of 3^n under the mapping g(x)->(1/(c(x)sqrt(1-4x))g(xc(x)).
Hankel transform is A088138(n+1). - Paul Barry (pbarry(AT)wit.ie), Jan 11 2007
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FORMULA
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G.f.: (sqrt(1-4x)+1)/(sqrt(1-4x)(3sqrt(1-4x)-1)); G.f.: sqrt(1-4x)(sqrt(1-4x)-3x+1)/((1-4x)(2-9x)); a(n)=sum{k=0..n, binomial(2n, n-k)2^k}.
a(n)=sum{k=0..n, C(2n,k)*2^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), Jan 11 2007
a(n)=sum{k=0..n, C(n+k-1,k)3^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), Sep 28 2007
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CROSSREFS
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Cf. A032443.
Sequence in context: A104631 A106391 A063881 this_sequence A052913 A129160 A143646
Adjacent sequences: A100189 A100190 A100191 this_sequence A100193 A100194 A100195
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Nov 08 2004
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