A103211 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*3^i*4^(n-i), a(0)=1.

1, 4, 28, 244, 2380, 24868, 272188, 3080596, 35758828, 423373636, 5092965724, 62071299892, 764811509644, 9511373563492, 119231457692284, 1505021128450516, 19112961439180588, 244028820862442116, 3130592301487969948, 40333745806536135028, 521655330655122923980

Offset: 0

Author

Ralf Stephan, Jan 27 2005

Keywords

• nonn

Comments

The Hankel transform of this sequence is 12^C(n+1,2). - Philippe Deléham, Oct 28 2007

The sequence 1, 1, 4, 28, ... has a(n) = 0^n + Sum_{k=0..n-1} C(n+k-1, 2*k)*C(k)*3^k and Hankel transform 3^C(n+1, 2)*4^C(n, 2). - Paul Barry, Dec 09 2008

Number of Dyck n-paths with two colors of up (U,u) and two colors of down (D,d) avoiding DU. - David Scambler, Jun 24 2013

Examples

			G.f. = 1 + 4*x + 28*x^2 + 244*x^3 + 2380*x^4 + 24868*x^5 + ... _Michael Somos_, Mar 15 2024
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, b_n(3).
• 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
• Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
• Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], (2016), eq. (1.13), a=4, b=3.
• 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.
• Djamila Oudrar, Sur l'énumération de structures discrètes, une approche par la théorie des relations, Thesis (in French), arXiv:1604.05839 [math.CO], 2016.

Crossrefs

Fourth column of array A103209.

Cf. A131763.

Programs

GAP

a:=n->(1/n)*Sum([0..n-1],i->Binomial(n,i)*Binomial(n,i+1)*
3^i*4^(n-i));;
A103211:=Concatenation([1],List([1..20],n->a(n))); # Muniru A Asiru, Feb 11 2018

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-14*x+1))/(6*x))) // G. C. Greubel, Feb 10 2018

Maple

A103211_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1] + 3*add(a[j]*a[w-j-1], j=0..w-1) od;
convert(a, list) end: A103211_list(20); # Peter Luschny, Feb 29 2016

Mathematica

CoefficientList[Series[(1-x-Sqrt[x^2-14*x+1])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
a[n_] := Sum[(-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 4], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Jan 08 2018 *)
a[ n_] := If[n < 0, a[-1-n], SeriesCoefficient[2/(1 - x + Sqrt[1 - 14*x + x^2]), {x, 0, n}]]; (* Michael Somos, Mar 15 2024 *)

PARI

x='x+O('x^30); Vec((1-x-sqrt(x^2-14*x+1))/(6*x)) \\ G. C. Greubel, Feb 10 2018

PARI

{a(n) = if(n<0, a(-1-n), polcoeff(2/(1 - x + sqrt(1 - 14*x + x^2 + x*O(x^n))), n))}; /* Michael Somos, Mar 15 2024 */

Formula

G.f.: (1-z-sqrt(z^2-14*z+1))/(6*z).

a(n) = Sum_{k=0..n} C(n+k,2k)*3^k*C(k), C(n) given by A000108. - Paul Barry, May 21 2005

a(n) = Sum_{k=0..n} A060693(n,k)*3^(n-k). - Philippe Deléham, Apr 02 2007

a(0)=1, a(n) = a(n-1) + 3*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007

G.f.: 1/(1-x-3*x/(1-x-3*x/(1-x-3*x/(1-x-3*x/(1-... (continued fraction). - Paul Barry, Nov 07 2009

D-finite with recurrence: (n+1)*a(n) = 7*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ sqrt(24+14*sqrt(3))*(7+4*sqrt(3))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

a(n) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k)*hypergeom([k - n, n + 1], [k + 2], 4). - Peter Luschny, Jan 08 2018

G.f. A(x) satisfies: A(x) = (1 + 3*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020

From Michael Somos, Mar 15 2024: (Start)

a(n) = (4/3)*A131763(n) for n>0.

Given g.f. A(x) and y = -x*A(-x^2), then 3*y-1/y = x+1/x.

If a(n) := a(-1-n) for n<0, then 0 = a(n)*(+a(n+1) -35*a(n+2) +4*a(n+3)) +a(n+1)*(+7*a(n+1) +194*a(n+2) -35*a(n+3)) +a(n+2)*(+7*a(n+2) +a(n+3)) for all n in Z. (End)