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A101850
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A Catalan transform of Pell(n+1).
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5
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1, 2, 7, 26, 100, 392, 1555, 6218, 25006, 100988, 409162, 1661948, 6764194, 27575732, 112570675, 460058906, 1881978694, 7704907724, 31566153058, 129400608044, 530734613920, 2177792579072, 8939838222718, 36711025334948
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| A Catalan transform of the Pell numbers A000129(n+1) under the mapping G(x)-> G(xc(x)), c(x) the g.f. of A000108. The inverse mapping is H(x)->H(x(1-x)).
Hankel transform is 3^n. [From Paul Barry (pbarry(AT)wit.ie), Jan 19 2009]
Row sums of the Riordan matrix (1/(x+sqrt(1-4x)),(1-sqrt(1-4x))/(2(x+sqrt(1-4x))) (A188513) [Emanuele Munarini, Apr 2 2011]
Equals the INVERT transform of A026641: (1, 1, 4, 13, 46, 166,...). Example: a(4) = 100 = (1, 1, 2, 7, 26) dot (46, 13, 4, 1, 1) = (46 + 13 + 8 + 7 + 26 ) = 100. - Gary W. Adamson, Jan 10 2012
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..100
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FORMULA
| G.f.: 2/(3*sqrt(1-4*x)+2*x-1);
a(n)=sum{k=0..n, (k/(2*n-k))*binomial(2*n-k, n-k)*A000129(k+1)}.
a(n)=Sum_{k, 0<=k<=n} A039599(n,k)*A016116(k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2008]
G.f.: 1/(1-2x-3x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Jan 19 2009]
Contribution from Emanuele Munarini, Apr 2 2011 (Start)
a(n) = [x^n] (1-2*x)/((1-2*x-x^2)(1-x)^(n+1)).
a(n) = sum(binomial(2*n+1,n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2),k=0..n).
Recurrence: 2*(n+3)*a(n+3)-4*(4*n+9)*a(n+2)+(31*n+45)*a(n+1)+2*(2*n+3)*a(n). (End)
a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:
2, 3, 0, 0, 0, 0,...
1, 1, 1, 0, 0, 0,...
1, 1, 1, 1, 0, 0,...
1, 1, 1, 1, 1, 0,...
1, 1, 1, 1, 1, 1,...
...
- Gary W. Adamson, Jul 14 2011
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MATHEMATICA
| CoefficientList[Series[(1-2x+3Sqrt[1-4x])/(4-16x-2x^2), {x, 0, 24}], x] [Emanuele Munarini, Apr 2 2011]
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PROG
| (Maxima) makelist(sum(binomial(2*n+1, n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2), k, 0, n), n, 0, 12); [Emanuele Munarini, Apr 2 2011]
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CROSSREFS
| Cf. A081696, A026641, A188513.
Sequence in context: A126223 A114121 A049775 * A176280 A045868 A171711
Adjacent sequences: A101847 A101848 A101849 * A101851 A101852 A101853
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Dec 18 2004
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