A014431 -id:A014431 - cp's OEIS frontend

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

Clark Kimberling

nonn

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in two colors: red (R) and green (G) (i.e., Motzkin paths with the up steps in two colors). E.g., a(3)=7 because we have HHH, HRD, HGD, RDH, GDH, RHD and GHD. - Emeric Deutsch, Dec 25 2003
Equals inverse binomial transform of A071356: (1, 2, 6, 20, 72, ...). - Gary W. Adamson, Sep 03 2010
a(n) is the number of increasing unary-binary trees with associated permutation that avoids 231. For more information about increasing unary-binary trees with an associated permutation, see A245888. - Manda Riehl, Aug 07 2014

			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)

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.

Cf. A071356, A001003, A068764, A217275.

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 *)

{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 */

{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 */

{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 */

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

A014432 a(n) = Sum_{i=1..n-1} a(i)*a(n-1-i), with a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 3, 12, 30, 111, 363, 1353, 4917, 18777, 71769, 280506, 1103556, 4395009, 17622309, 71220828, 289510662, 1183627137, 4862148753, 20061888924, 83100910530, 345457823493, 1440734205513, 6026408186457, 25275954499905, 106277040064191

Offset: 0

Wouter Meeussen

nonn

Robert Israel, Table of n, a(n) for n = 0..1545

Cf. A025237.

seq(coeff(convert(series((1+x-sqrt(1-2*x-11*x^2))/(2*x),x,50),polynom),x,i),i=0..30);
A014431:=proc(n) options remember: local i: if n<2 then RETURN([1,3][n+1]) else RETURN(add(A014431(i)*A014431(n-1-i),i=1..n-1)) fi:end;seq(A014431(n),n=0..30); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

Rest[CoefficientList[Series[(1+x-Sqrt[1-2x-11x^2])/2,{x,0,30}],x]] (* Harvey P. Dale, Apr 17 2019 *)

a(n)=polcoeff((1+x-sqrt(1-2*x-11*x^2+x*O(x^n)))/2,n)

G.f.: ((1+x-sqrt(1-2*x-11*x^2)))/(2*x). - Michael Somos, Jun 08 2000; corrected by Robert Israel, Sep 10 2020
a(n) = (3/(11*n)) * ((3+n)*A025237(n+1) - (2*n+3)*A025237(n)) for n>0. [Mark van Hoeij, Jul 02 2010]
(n+1)*a(n) = (2*n-1)*a(n-1)+11*(n-2)*a(n-2). - Robert Israel, Sep 10 2020
G.f.: 1 + 3*x/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - ...))))) (continued fraction). - Nikolaos Pantelidis, Nov 24 2022

Corrected by C. Ronaldo (aga_new_ac(AT)hotmail.com) and Ralf Stephan, Dec 19 2004

A374487 Expansion of 1/(1 - 2x - 7x^2)^(3/2).

Original entry on oeis.org

1, 3, 18, 70, 315, 1281, 5348, 21708, 88245, 355135, 1425270, 5692050, 22666735, 89986365, 356400840, 1408459928, 5555679849, 21877337979, 86020384730, 337769595870, 1324677499299, 5189411915897, 20308936981932, 79406140870500, 310206869770525, 1210898719869111

Offset: 0

Seiichi Manyama, Jul 09 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A102839, A102840, A374488.

Cf. A014431, A025235, A084601.

Module[{x}, CoefficientList[Series[1/(1 - (7*x + 2)*x)^(3/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)

a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1));

a(0) = 1, a(1) = 3; a(n) = ((2*n+1)*a(n-1) + 7*(n+1)*a(n-2))/n.
a(n) = binomial(n+2,2) * A025235(n).
From Seiichi Manyama, Aug 20 2025: (Start)
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 2^k * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (1/2)^k * (7/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)
a(n) ~ sqrt(n) * (1 + 2*sqrt(2))^(n + 3/2) / (2^(11/4) * sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

cp's OEIS Frontend

A025235 a(n) = (1/2)*s(n+2), where s = A014431.

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A014432 a(n) = Sum_{i=1..n-1} a(i)*a(n-1-i), with a(0) = 1, a(1) = 3.

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A374487 Expansion of 1/(1 - 2x - 7x^2)^(3/2).

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cp's OEIS Frontend

A025235 a(n) = (1/2)*s(n+2), where s = A014431.

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A014432 a(n) = Sum_{i=1..n-1} a(i)*a(n-1-i), with a(0) = 1, a(1) = 3.

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A374487 Expansion of 1/(1 - 2*x - 7*x^2)^(3/2).

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A374487 Expansion of 1/(1 - 2x - 7x^2)^(3/2).