A025235 a(n) = (1/2)*s(n+2), where s = A014431.
1, 1, 3, 7, 21, 61, 191, 603, 1961, 6457, 21595, 72975, 249085, 857013, 2970007, 10356323, 36311633, 127937649, 452738867, 1608426647, 5734534629, 20511509549, 73583105007, 264687136235, 954482676217, 3449853902761, 12495597328011, 45349353908383
Offset: 0
Keywords
Examples
x + x^2 + 3*x^3 + 7*x^4 + 21*x^5 + 61*x^6 + 191*x^7 + 603*x^8 + 1961*x^9 + ... a(4) = 21 since the top row of M^4 = (21, 11, 7, 1, 1)
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
- Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
- Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
- Stefano Capparelli and Alberto Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, 18 (2015), #15.8.5.
- Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
- Serkan Demiriz, Adem Şahin, and Sezer Erdem, Some topological and geometric properties of novel generalized Motzkin sequence spaces, Rendiconti Circ. Mat. Palermo Ser. 2 (2025) Vol. 74, No. 136. See p. 4.
- Maciej Dziemiańczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.
- Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
- Louis W. Shapiro and Carol J. Wang, A bijection between 3-Motzkin paths and Schroder paths with no peak at odd height, JIS 12 (2009) 09.3.2.
Programs
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Mathematica
Join[{1}, Table[Sum[2^(k - 1)*Binomial[n + 1, k]*Binomial[n - k + 1, k - 1]/(n + 1), {k,0,n}], {n,0,50}]] (* G. C. Greubel, Jan 27 2017 *) a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 2, 8]; Table[a[n], {n, 0, 27}] (* Peter Luschny, Mar 18 2018 *) -
PARI
{a(n) = if( n<0, 0, polcoeff( serreverse( x / (1 + x + 2*x^2 + x * O(x^n))), n+1))} /* Michael Somos, Jul 12 2003 */ -
PARI
{a(n) = if( n<0, 0, polcoeff( (1 - x - sqrt(1 - 2*x -7*x^2 + x^3 * O(x^n)) ) / 4, n+2))} /* Michael Somos, Mar 31 2007 */ -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * simplify( polcoeff( exp(x + A) * besseli(1, 2*x * quadgen(8) + A), n)))} /* Michael Somos, Mar 31 2007 */
Formula
a(n) = Sum_{k=0..n} 2^(k-1)*binomial(n+1, k)*binomial(n-k+1, k-1)/(n+1 ). - Len Smiley
G.f.: (1 - x - sqrt(1 - 2*x - 7*x^2)) / (4*x^2). - Michael Somos, Jun 08 2000
G.f. (for offset 1) is series reversion of x / (1 + x + 2*x^2). - Michael Somos, Jul 12 2003
a(n) = Sum_{k=0..n} binomial(n, k)*2^(k/2)*C(k/2)*(1+(-1)^k)/2, where C(n)=A000108(n). - Paul Barry, Dec 22 2003
E.g.f.: exp(x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic, Mar 31 2004
From Gary W. Adamson, Feb 21 2012: (Start)
a(n) is the leftmost term in the top row of M^n, M is an infinite square production matrix as follows:
1, 1, 0, 0, 0, 0, ...
2, 0, 1, 0, 0, 0, ...
2, 2, 0, 1, 0, 0, ...
2, 2, 2, 0, 1, 0, ...
2, 2, 2, 2, 0, 1, ...
2, 2, 2, 2, 2, 0, ...
2, 2, 2, 2, 2, 2, ...
... (End)
From Vaclav Kotesovec, Sep 29 2012: (Start)
a(n) ~ (1+2*sqrt(2))^(n+3/2)/(2*sqrt(Pi)*2^(3/4)*n^(3/2)).
Recurrence: (n+2)*a(n) = (2*n+1)*a(n-1) + 7*(n-1)*a(n-2). (End)
a(n) = hypergeom([-n/2, (1-n)/2], [2], 8). - Peter Luschny, May 28 2014
G.f.: 1/(1 - x - 2*x^2/(1 - x - 2*x^2/(1 - x - 2*x^2/(1 - x - 2*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017
Comments