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A103210 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*2^i*3^(n-i), a(0)=1. 19
1, 3, 15, 93, 645, 4791, 37275, 299865, 2474025, 20819307, 178003815, 1541918901, 13503125805, 119352115551, 1063366539315, 9539785668657, 86104685123025, 781343125570515, 7124072211203775, 65233526296899981, 599633539433039445, 5531156299278726663 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 28 2007

The Hankel transform of the sequence starting 1, 1, 3, 15, ... is A081955. - Paul Barry, Dec 09 2008

Number of Schroeder paths from (0,0) to (0,2n) allowing two colors for the down steps (or alternatively for the rise steps). - Paul Barry, Feb 01 2009

Essentially reversion of x*(1-2*x)/(1+x). - Paul Barry, Apr 28 2009

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

J. Abate, W. Whitt, Integer Sequences from Queueing Theory , J. Int. Seq. 13 (2010), 10.5.5. b_n(2).

E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett. 98 (4) (2006) 162-167

Z. Chen, H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016, eq. (1.13), a=3, b=2.

Samuele Giraudo, Operads from posets and Koszul duality, arXiv preprint arXiv:1504.04529 [math.CO], 2015.

Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.

FORMULA

G.f.: (1-z-sqrt(z^2-10*z+1))/(4*z).

a(n) = Sum_{k=0..n} C(n+k, 2k)2^k*C(k), C(n) given by A000108. - Paul Barry, May 21 2005

a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k). - Philippe Deléham, Apr 02 2007

a(0) = 1, a(n) = a(n-1) + 2*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007

a(n) = (3/2)*A107841(n) for n > 0. - Philippe Deléham, Oct 28 2007

G.f.: 1/(1-x-2x/(1-x-2x/(1-x-2x/(1-.... (continued fraction). - Paul Barry, Feb 01 2009

G.f.: 1/(1-3x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-... (continued fraction). - Paul Barry, Apr 28 2009

G.f.: 1/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x/(1-... (continued fraction). - Paul Barry, May 14 2009

a(n) = Hypergeometric2F1(-n,n+1,2,-2) = sum{k=0..n, C(n+k,k) * C(n,k) * 2^k/(k+1)}. - Paul Barry, Feb 08 2011

G.f.: A(x) = (1-x-(x^2-10*x+1)^(1/2))/(4*x) = 1/(G(0)-x); G(k)= 1 + x - 3*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012

Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

MAPLE

A103210 := proc(n)

    if n = 0 then

        1;

    else

        add(binomial(n, i)*binomial(n, i+1)*2^i*3^(n-i), i=0..n-1)/n ;

    end if;

end proc: # R. J. Mathar, Feb 10 2015

A103210_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

for w from 1 to n do a[w] := 3*a[w-1] + 2*add(a[j]*a[w-j-1], j=1..w-1) od;

convert(a, list) end: A103210_list(21); # Peter Luschny, Feb 29 2016

MATHEMATICA

CoefficientList[Series[(1-x-Sqrt[x^2-10*x+1])/(4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

A103210[n_] := Hypergeometric2F1[-n, n + 1, 2, -2]; Table[A103210[n], {n, 0, 21}] (* Peter Luschny, Jan 07 2018 *)

PROG

(PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-10*x+1))/(4*x)) \\ G. C. Greubel, Feb 10 2018

(MAGMA) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-10*x+1))/(4*x))) // G. C. Greubel, Feb 10 2018

CROSSREFS

Third column of array A103209.

Sequence in context: A256335 A258313 A074539 * A203014 A060066 A206177

Adjacent sequences:  A103207 A103208 A103209 * A103211 A103212 A103213

KEYWORD

nonn

AUTHOR

Ralf Stephan, Jan 27 2005

EXTENSIONS

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

STATUS

approved

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Last modified October 14 03:58 EDT 2019. Contains 327995 sequences. (Running on oeis4.)