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.

1, 3, 15, 93, 645, 4791, 37275, 299865, 2474025, 20819307, 178003815, 1541918901, 13503125805, 119352115551, 1063366539315, 9539785668657, 86104685123025, 781343125570515, 7124072211203775, 65233526296899981, 599633539433039445, 5531156299278726663

Offset: 0

Author

Ralf Stephan, Jan 27 2005

Keywords

• nonn

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

a(n) is also the number of infix expressions with n variables and operators +, - and * (or +, * and /) such that there are no redundant parentheses. - Vjeran Crnjak, Apr 25 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Joseph Abate and Ward Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5. b_n(2).
• Eyal Ackerman, Gill Barequet, Ron Y. Pinter, and Dan Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett. (2006) Vol. 98, No. 4, 162-167.
• Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
• Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
• Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016, eq. (1.13), a=3, b=2.
• Robert Dickau, 3D Guillotine Partitions, figures for 3D slices.
• 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.
• Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Crossrefs

Third column of array A103209.

Cf. A000108, A060693, A081955, A107841.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!((1-x-Sqrt(x^2-10*x+1))/(4*x))); // G. C. Greubel, Feb 10 2018

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, 25}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
A103210[n_]:= Hypergeometric2F1[-n, n+1, 2, -2]; Table[A103210[n], {n, 0, 25}] (* Peter Luschny, Jan 07 2018 *)

PARI

my(x='x+O('x^25)); Vec((1-x-sqrt(x^2-10*x+1))/(4*x)) \\ G. C. Greubel, Feb 10 2018

Sage

[1]+[(3^n/n)*sum( binomial(n,j)*binomial(n,j+1)*(2/3)^j for j in (0..n-1)) for n in (1..25)] # G. C. Greubel, Jun 08 2020

Formula

G.f.: (1 - z - sqrt(1 -10*z +z^2))/(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-2*x/(1-x-2*x/(1-x-2*x/(1-.... (continued fraction). - Paul Barry, Feb 01 2009

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

G.f.: 1/(1-3*x/(1-2*x/(1-3*x/(1-2*x/(1-3*x/(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

D-finite with 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

G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020

Extensions

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