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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(xc(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

REFERENCES

Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From N. J. A. Sloane, Oct 08 2012

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

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). - Philippe Deléham, 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). - 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(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

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]

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]

CROSSREFS

Cf. A081696, A026641, A188513.

Sequence in context: A273320 A114121 A049775 * A279002 A176280 A045868

Adjacent sequences:  A101847 A101848 A101849 * A101851 A101852 A101853

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Dec 18 2004

STATUS

approved

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Last modified February 24 06:13 EST 2018. Contains 299597 sequences. (Running on oeis4.)