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
Keywords
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.
Programs
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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
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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
Comments