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A101850 A Catalan transform of Pell(n+1). 5
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; text; internal format)
OFFSET
0,2
COMMENTS
A Catalan transform of the Pell numbers A000129(n+1) under the mapping G(x) -> G(x*c(x)), c(x) the g.f. of A000108. The inverse mapping is H(x) -> H(x*(1-x)).
Hankel transform is 3^n. - Paul Barry, 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 02 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
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. [N. J. A. Sloane, Oct 08 2012]
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..n} A039599(n,k)*A016116(k). - Philippe Deléham, Oct 29 2008
G.f.: 1/(1-2*x-3*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-... (continued fraction). - Paul Barry, Jan 19 2009
From Emanuele Munarini, Apr 02 2011: (Start)
a(n) = [x^n] (1-2*x)/((1-2*x-x^2)(1-x)^(n+1)).
a(n) = Sum_{k=0..n} binomial(2*n+1, n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2).
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) = 0. (End)
From Gary W. Adamson, Jul 14 2011: (Start)
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, ...
... (End)
a(n) ~ (1/4)*(2 + 3/sqrt(2))^n. - Vaclav Kotesovec, Oct 17 2012
MATHEMATICA
CoefficientList[Series[(1-2x+3Sqrt[1-4x])/(4-16x-2x^2), {x, 0, 24}], x] (* Emanuele Munarini, Apr 02 2011 *)
CoefficientList[Series[2/(3*Sqrt[1-4*x]+2*x-1), {x, 0, 30}], x] (* or *) RecurrenceTable[{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]==0, a[0]==1, a[1]==2, a[2]==7}, a, {n, 0, 30}](* G. C. Greubel, Feb 18 2019 *)
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 02 2011 */
(PARI) my(x='x+O('x^30)); Vec(2/(3*sqrt(1-4*x)+2*x-1)) \\ G. C. Greubel, Feb 18 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2/(3*Sqrt(1-4*x)+2*x-1) )); // G. C. Greubel, Feb 18 2019
(Sage) (2/(3*sqrt(1-4*x)+2*x-1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019
CROSSREFS
Sequence in context: A273320 A114121 A049775 * A279002 A176280 A349713
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 18 2004
STATUS
approved

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Last modified March 19 06:56 EDT 2024. Contains 370953 sequences. (Running on oeis4.)