A025235 -id:A025235 - cp's OEIS frontend

A071356 Expansion of (1 - 2x - sqrt(1 - 4x - 4x^2))/(4x^2).

1, 2, 6, 20, 72, 272, 1064, 4272, 17504, 72896, 307648, 1312896, 5655808, 24562176, 107419264, 472675072, 2091206144, 9296612352, 41507566592, 186045061120, 836830457856, 3776131489792, 17089399689216, 77548125675520, 352766964908032

Offset: 0

N. J. A. Sloane, Jun 12 2002

nonn

Number of underdiagonal lattice paths from (0,0) to the line x=n, using only steps R=(1,0), V=(0,1) and D=(1,2). Also number of Motzkin paths of length n in which both the "up" and the "level" steps come in two colors. E.g., a(2)=6 because we have RR, RVR, RRV, RD, RVRV and RRVV. - Emeric Deutsch, Dec 21 2003
Inverse binomial transform of little Schroeder numbers 1,3,11,... (A001003 with first term deleted). - David Callan, Feb 07 2004
a(n) is the number of planar trees satisfying: 1) Every internal node has at least two children, 2) Among the children of a node, only the leftmost and the rightmost children can be leaves, 3) The tree has n+1 leaves. For instance, a(3)=6. - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005
Hankel transform is A006125(n+1)=2^C(n+1,2). - Paul Barry, Jan 08 2008
Equals binomial transform of A025235: (1, 1, 3, 7, 21, 61, 191, ...). - Gary W. Adamson, Sep 03 2010
Conjecturally, the number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) > e(j) <= e(k). [Martinez and Savage, 2.19] - Eric M. Schmidt, Jul 17 2017
Let s denote West's stack-sorting map, and let Av_n(tau_1, ..., tau_r) denote the set of permutations of [n] that avoid the patterns tau_1, ..., tau_r. It is conjectured that a(n) = |s^{-1}(Av_{n+1}(132, 231))| = |s^{-1}(Av_{n+1}(132, 312))| = |s^{-1}(Av_{n+1}(231, 312))|. Only the last of these equalities is known. - Colin Defant, Sep 16 2018

			a(3) = 20 = sum of top row terms in M^3 = (9 + 7 + 3 + 1).

Fung Lam, Table of n, a(n) for n = 0..1465
Marcelo Aguiar and Walter Moreira, Combinatorics of the free Baxter algebra, arXiv:math/0510169 [math.CO], 2005-2007, see Corollary 3.3.iii.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Miklos Bona, Stack-sorting preimages and 0-1-trees, arXiv:2505.18295 [math.CO], 2025. See p. 3.
Vuong Bui, Bounding Klarner's constant from above using a simple recurrence, arXiv:2412.20143 [math.CO], 2024.
Wenqin Cao, Emma Yu Jin, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Colin Defant, Enumeration of Stack-Sorting Preimages via a Decomposition Lemma, arXiv:1904.02829 [math.CO], 2019.
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.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
Li Guo and Yunnan Li, Braided dendriform and tridendriform algebras and braided Hopf algebras of planar trees, arXiv:1906.06454 [math.QA], 2019.
Bin Han, The gamma-positive coefficients arising in segmented permutations, Discrete Math., 344 (2012), #112336. See p. 7.
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.
Chetak Hossain, Quotients Derived from Posets in Algebraic and Topological Combinatorics, Ph. D. Dissertation, North Carolina State University (2019).
Germain Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 32-33 (same sequence but with offset 1).
Germain Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.
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.

A036774(n) = a(n-1) * n! / 2^(n-1).
Row sums of A071943.
Cf. A000108, A025235.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!((1 - 2*x - Sqrt(1 - 4*x - 4*x^2))/(4*x^2))); // Vincenzo Librandi, Jan 21 2020

CoefficientList[Series[(1-2*x-Sqrt[1-4*x-4*x^2])/(4*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 24 2013 *)
a[n_] := 2^n Hypergeometric2F1[(1-n)/2, -n/2, 2, 2];
Table[a[n], {n, 0, 24}] (* Peter Luschny, May 30 2021 *)

a(n)=if(n<0,0,n++; polcoeff(serreverse(x/(1+2*x+2*x^2)+x*O(x^n)),n))

{a(n)= if(n<1, n==0, polcoeff( 2/(1 -2*x +sqrt(1 -4*x -4*x^2 +x*O(x^n))), n))}

{a(n)= local(A); if(n<0, 0, A= x*O(x^n); n!*simplify(polcoeff( exp(2*x +A)* besseli(1, 2*x* quadgen(8) +A), n)))} /* Michael Somos, Mar 31 2007 */

def A071356_list(n):  # n>=1
    T = [0]*(n+1); R = [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a + 2*(b + c)
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u; R.append(u)
    return R
A071356_list(25)  # Peter Luschny, Nov 01 2012

G.f. A(x) satisfies 2x^2*A(x)^2+(2x-1)*A(x)+1=0 and A(x)=1/(1-2x-2x^2/A(x)). - Michael Somos, Sep 06 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)C(k)2^(n-2k)*2^k. - Paul Barry, May 18 2005
G.f.: (1 - 2*x - sqrt(1 - 4*x - 4*x^2) )/(4*x^2) = 2/(1 - 2*x +sqrt(1 - 4*x - 4*x^2)).
Moment representation is a(n) = (1/(4*Pi))*int(x^n*sqrt(4-4x-x^2), x, -2*sqrt(2)-2, 2*sqrt(2)-2). - Paul Barry, Jan 08 2008
G.f.: 1/(1-2x-2x^2/(1-2x-2x^2/(1-2x-2x^2/(1-2x-2x^2/(1-2x-2x^2/.... (continued fraction). - Paul Barry, Dec 06 2008
From Gary W. Adamson, Jul 22 2011: (Start)
a(n) = sum of top row terms of M^n, M = an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  2, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  2, 2, 2, 1, 1, 0, ...
  2, 2, 2, 2, 1, 1, ...
  2, 2, 2, 2, 2, 1, ... (End)
E.g.f.: a(n) = n!* [x^n] exp(2*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Peter Luschny, Aug 25 2012
D-finite with recurrence: (n+2)*a(n) +2*(-2*n-1)*a(n-1) +4*(-n+1)*a(n-2)=0. - R. J. Mathar, Dec 02 2012 (Formula verified and used for computations. - Fung Lam, Feb 24 2014)
a(n) ~ 2^(n - 1/4) * (1+sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 24 2013, simplified Jan 26 2019
a(n) = A179190(n+2)/4. - R. J. Mathar, Jan 20 2020
a(n) = 2^n * hypergeom((1 - n)/2, -n/2, 2, 2). - Peter Luschny, May 30 2021
a(n) = (-2*î)^(n+2) * (Legendre_P(n+2, i) - Legendre_P(n, i))/(4*(2*n + 3)). - Peter Bala, May 06 2024
From Emanuele Munarini, Jun 13 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*2^(n-k)/(k+1).
a(n) = Sum_{k=0..floor((n+2)/3)} binomial(n-2k+2, 2k)*Catalan(n-2k+1).
a(n) = Sum_{k=0..floor((n+2)/4)} binomial(n-2k+1, 2k+1)*Catalan(n-2k). (End)

A068764 Generalized Catalan numbers 2xA(x)^2 -A(x) +1 -x =0.

Original entry on oeis.org

1, 1, 4, 18, 88, 456, 2464, 13736, 78432, 456416, 2697088, 16141120, 97632000, 595912960, 3665728512, 22703097472, 141448381952, 885934151168, 5575020435456, 35230798994432, 223485795258368, 1422572226146304, 9083682419818496, 58169612565614592, 373486362257899520, 2403850703479816192

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(2,2; n)/2 with K(a,b; n) defined in a comment to A068763.

Hankel transform is A166232(n+1). - Paul Barry, Oct 09 2009

			G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 456*x^5 + 2464*x^6 + 13736*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A025227, A025228, A025229, A025230.

Cf. A071356, A001003, A025235.

Table[SeriesCoefficient[(1-Sqrt[1-8*x*(1-x)])/(4*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
Round@Table[4^(n-1) Hypergeometric2F1[(1-n)/2, 1-n/2, 2, 1/2] + KroneckerDelta[n]/Sqrt[2], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4^(n - 1) Hypergeometric2F1[ (1 - n)/2, (2 - n)/2, 2, 1/2]]; (* Michael Somos, Nov 08 2015 *)

a(n):=sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n); /* Vladimir Kruchinin, Aug 11 2010 */

{a(n) = my(A); if( n<1, n==0, n--;  A = x * O(x^n); n! * simplify( polcoeff( exp(4*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* Michael Somos, Mar 31 2007 */

x='x+O('x^66); Vec((1-sqrt(1-8*x*(1-x)))/(4*x)) \\ Joerg Arndt, May 06 2013

G.f.: (1-sqrt(1-8*x*(1-x)))/(4*x).
a(n+1) = 2*sum(a(k)*a(n-k), k=0..n), n>=1, a(0) = 1 = a(1).
a(n) = (2^n)*p(n, -1/2) with the row polynomials p(n, x) defined from array A068763.
E.g.f. (offset -1) is exp(4*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic, Mar 31 2004
The o.g.f. satisfies A(x) = 1 + x*(2*A(x)^2 - 1), A(0) = 1. - Wolfdieter Lang, Nov 13 2007
a(n) = subs(t=1,(d^(n-1)/dt^(n-1))(-1+2*t^2)^n)/n!, n >= 2, due to the Lagrange series for the given implicit o.g.f. equation. This formula holds also for n=1 if no differentiation is used. - Wolfdieter Lang, Nov 13 2007, Feb 22 2008
1/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-..... (continued fraction). - Paul Barry, Jan 29 2009
a(n) = A166229(n)/(2-0^n). - Paul Barry, Oct 09 2009
a(n) = sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n), n>0. - Vladimir Kruchinin, Aug 11 2010
D-finite with recurrence: (n+1)*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
a(n) = 4^(n-1)*hypergeom([(1-n)/2,1-n/2], [2], 1/2) + 0^n/sqrt(2). - Vladimir Reshetnikov, Nov 07 2015
0 = a(n)*(+64*a(n+1) - 160*a(n+2) + 32*a(n+3)) + a(n+1)*(+32*a(n+1) + 48*a(n+2) - 20*a(n+3)) + a(n+2)*(+4*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Nov 08 2015
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(2*k+1,n) / (2*k+1). - Seiichi Manyama, Jul 24 2023

A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

Original entry on oeis.org

1, 1, 8, 22, 141, 561, 3291, 15583, 88691, 459187, 2599570, 14136200, 80391235, 450046143, 2579291352, 14710321998, 85002979083, 491050703739, 2859262171872, 16674374605722, 97747766045679, 574231140306699, 3385974360904227, 20009363692187115, 118582649963026677

Offset: 0

Vaclav Kotesovec, Sep 29 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001006 (z=1), A025235 (z=2), A025237 (z=3), A091147 (z=4), A091148 (z=5), A091149 (z=6).

Table[SeriesCoefficient[2/(1-x+Sqrt[1-2*x-27*x^2]),{x,0,n}],{n,0,25}]
Table[Sum[Binomial[n,2k]*Binomial[2k,k]*7^k/(k+1),{k,0,n}],{n,0,25}]

Generally for G.f. = 2/(1-x+sqrt(1-2x-(4*z-1)*x^2)) is asymptotic
a(n) ~ (1+2*sqrt(z))^(n+3/2)/(2*sqrt(Pi)*z^(3/4)*n^(3/2)); here we have the case z=7.
D-finite with recurrence: (n+2)*a(n)=(2*n+1)*a(n-1)+(4*z-1)*(n-1)*a(n-2);; here with z=7.
G.f.: 1/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017

A014431 a(1) = 1, a(2) = 2, a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-2)*a(2) for n >= 3.

Original entry on oeis.org

1, 2, 2, 6, 14, 42, 122, 382, 1206, 3922, 12914, 43190, 145950, 498170, 1714026, 5940014, 20712646, 72623266, 255875298, 905477734, 3216853294, 11469069258, 41023019098, 147166210014, 529374272470, 1908965352434, 6899707805522, 24991194656022, 90698707816766

Offset: 1

Clark Kimberling and Wouter Meeussen

Michael De Vlieger, Table of n, a(n) for n = 1..1724
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Cf. A025235, A006318, A084601.

a:=[1,2]; for n in [3..30] do Append(~a,&+[a[k]*a[n-k]:k in [1..n-2]] ); end for; a; // Marius A. Burtea, Jan 02 2020

Rest@ CoefficientList[Series[(1 + x - Sqrt[1 - 2 x - 7 x^2])/2, {x, 0, 27}], x] (* Michael De Vlieger, Jan 02 2020 *)

a(n)=polcoeff((1+x-sqrt(1-2*x-7*x^2+x*O(x^n)))/2,n)

a(n) = 2*A025235(n-2) for n>=2.
G.f.: (1+x-sqrt(1-2*x-7*x^2))/2. - Michael Somos, Jun 08 2000
a(n) = (A084601(n) - A084601(n-1))/(2*(n-1)) for n > 1. - Mark van Hoeij, Jul 02 2010
G.f.: x + 2*x^2/G(0) with G(k) = (1 - x - 2*x^2/G(k+1)) (continued fraction). - Nikolaos Pantelidis, Dec 16 2022
From Peter Bala, May 01 2024: (Start)
O.g.f.: A(x) = x*S(x/(1 + 2*x)) = 2*x - x*S(- x/(1 - 4*x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318.
The g.f. satisfies A(x)^2 - (1 + x)*A(x) + x*(1 + 2*x) = 0.
A(x) = x*(1 + 2*x)/(1 + x - x*(1 + 2*x)/(1 + x - x*(1 + 2*x)/(1 + x - ...))).
A(x) = x/(1 - 2*x/(1 + 2*x - x/(1 - 2*x/(1 + 2*x - x/(1 - 2*x/(1 + 2*x - x/(1 - ...))))))). (End)
D-finite with recurrence n*a(n) +(-2*n+3)*a(n-1) +7*(-n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

Corrected by T. D. Noe, Oct 31 2006

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 9, 1, 1, 1, 5, 10, 21, 21, 1, 1, 1, 6, 13, 37, 61, 51, 1, 1, 1, 7, 16, 57, 121, 191, 127, 1, 1, 1, 8, 19, 81, 201, 451, 603, 323, 1, 1, 1, 9, 22, 109, 301, 861, 1639, 1961, 835, 1

Offset: 0

Seiichi Manyama, May 06 2019

			Square array begins:
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   2,   3,    4,    5,    6,     7,     8, ...
   1,   4,   7,   10,   13,   16,    19,    22, ...
   1,   9,  21,   37,   57,   81,   109,   141, ...
   1,  21,  61,  121,  201,  301,   421,   561, ...
   1,  51, 191,  451,  861, 1451,  2251,  3291, ...
   1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..7 give A000012, A001006, A025235, A025237, A091147, A091148, A091149, A217275.
Main diagonal gives A307906.
Cf. A000108, A107267, A247495, A307855.

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 + x + k*x^2)^(n+1).
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * A000108(j).
(n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

A359364 Triangle read by rows. The Motzkin triangle, the coefficients of the Motzkin polynomials. M(n, k) = binomial(n, k) * CatalanNumber(k/2) if k is even, otherwise 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 2, 1, 0, 10, 0, 10, 0, 1, 0, 15, 0, 30, 0, 5, 1, 0, 21, 0, 70, 0, 35, 0, 1, 0, 28, 0, 140, 0, 140, 0, 14, 1, 0, 36, 0, 252, 0, 420, 0, 126, 0, 1, 0, 45, 0, 420, 0, 1050, 0, 630, 0, 42, 1, 0, 55, 0, 660, 0, 2310, 0, 2310, 0, 462, 0

Offset: 0

Peter Luschny, Jan 09 2023

The generalized Motzkin numbers M(n, k) are a refinement of the Motzkin numbers M(n) (A001006) in the sense that they are coefficients of polynomials M(n, x) = Sum_{n..k} M(n, k) * x^k that take the value M(n) at x = 1. The coefficients of x^n are the aerated Catalan numbers A126120.
Variants are the irregular triangle A055151 with zeros deleted, A097610 with reversed rows, A107131 and A080159.
In the literature the name 'Motzkin triangle' is also used for the triangle A026300, which is generated from the powers of the generating function of the Motzkin numbers.

			Triangle M(n, k) starts:
[0] 1;
[1] 1, 0;
[2] 1, 0,  1;
[3] 1, 0,  3, 0;
[4] 1, 0,  6, 0,   2;
[5] 1, 0, 10, 0,  10, 0;
[6] 1, 0, 15, 0,  30, 0,   5;
[7] 1, 0, 21, 0,  70, 0,  35, 0;
[8] 1, 0, 28, 0, 140, 0, 140, 0,  14;
[9] 1, 0, 36, 0, 252, 0, 420, 0, 126, 0;

M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017, (Table 1 on p. 3).

Variants: A055151, A080159, A097610, A107131.
Columns M(n, 2*k): A000012, A000217, A034827, A000910, A088625, A088626, A213380.
Cf. A001006 (Motzkin numbers), A026300 (Motzkin gf. triangle), A126120 (aerated Catalan), A000108 (Catalan).
Cf. A138364, A107587, A002457, A002522, A025179, A025235, A056107, A014531, A023426, A091147, A189912, A343386, A343773, A359647, A359649.

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
M := (n, k) -> ifelse(irem(k, 2) = 1, 0, CatalanNumber(k/2)*binomial(n, k)):
for n from 0 to 9 do seq(M(n, k), k = 0..n) od;
# Alternative, as coefficients of polynomials:
p := n -> hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2):
seq(print(seq(coeff(simplify(p(n)), x, k), k = 0..n)), n = 0..9);
# Using the exponential generating function:
egf := exp(x)*BesselI(1, 2*x*t)/(x*t): ser:= series(egf, x, 11):
seq(print(seq(coeff(simplify(n!*coeff(ser, x, n)), t, k), k = 0..n)), n = 0..9);

from functools import cache
@cache
def M(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return (2 * (n - 1) * M(n - 2, n - 2)) // (n // 2 + 1)
    return (M(n - 1, k) * n) // (n - k)
for n in range(10): print([M(n, k) for k in range(n + 1)])

Let p(n, x) = hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2).
p(n, 1) = A001006(n); p(n, sqrt(2)) = A025235(n); p(n, 2) = A091147(n).
p(2, n) = A002522(n); p(3, n) = A056107(n).
p(n, n) = A359649(n); 2^n*p(n, 1/2) = A000108(n+1).
M(n, k) = [x^k] p(n, x).
M(n, k) = [t^k] (n! * [x^n] exp(x) * BesselI(1, 2*t*x) / (t*x)).
M(n, k) = [t^k][x^n] ((1 - x - sqrt((x-1)^2 - (2*t*x)^2)) / (2*(t*x)^2)).
M(n, n) = A126120(n).
M(n, n-1) = A138364(n), the number of Motzkin n-paths with exactly one flat step.
M(2*n, 2*n) = A000108(n), the number of peakless Motzkin paths having a total of n up and level steps.
M(4*n, 2*n) = A359647(n), the central terms without zeros.
M(2*n+2, 2*n) = A002457(n) = (-4)^n * binomial(-3/2, n).
Sum_{k=0..n} M(n - k, k) = A023426(n).
Sum_{k=0..n} k * M(n, k) = 2*A014531(n-1) = 2*GegenbauerC(n - 2, -n, -1/2).
Sum_{k=0..n} i^k*M(n, k) = A343773(n), (i the imaginary unit), is the excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).
Sum_{k=0..n} Sum_{j=0..k} M(n, j) = A189912(n).
Sum_{k=0..n} Sum_{j=0..k} M(n, n-j) = modified A025179(n).
For a recursion see the Python program.

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

A294642 a(n) = n! * [x^n] exp(nx)BesselI(1,2sqrt(2)x)/(sqrt(2)*x).

Original entry on oeis.org

1, 1, 6, 45, 456, 5825, 89896, 1627437, 33822944, 793783233, 20765009344, 599157626925, 18904594000128, 647524807918209, 23929038677825152, 948995910652193325, 40203601321988822528, 1812025020244371552897, 86577002960871477916672, 4371100278517527047687213

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000108, A001003, A025235, A068764, A071356, A151374, A247496, A292716.

Simplify[Table[n! SeriesCoefficient[Exp[n x] BesselI[1, 2 Sqrt[2] x]/(Sqrt[2] x), {x, 0, n}], {n, 0, 19}]]
Table[SeriesCoefficient[(1 - n x - Sqrt[1 - 2 n x + (n^2 - 8) x^2])/(4 x^2), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-2 x^2, 1 - n x, {i, 1, n}]), {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[2^k n^(n - 2 k) Binomial[n, 2 k] CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 1, 19}]]
Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 8/n^2], {n, 1, 19}]]

a(n) = [x^n] (1 - n*x - sqrt(1 - 2*n*x + (n^2 - 8)*x^2))/(4*x^2).
a(n) = [x^n] 1/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - ...))))), a continued fraction.
a(n) = Sum_{k=0..floor(n/2)} 2^k*n^(n-2*k)*binomial(n,2*k)*A000108(k).
a(n) = n^n*2F1(1/2-n/2,-n/2; 2; 8/n^2).
a(n) ~ c * n^n, where c = BesselI(1, 2*sqrt(2))/sqrt(2) = 2.3948330992734... - Vaclav Kotesovec, Nov 06 2017

A348202 Number of nonnegative lattice paths from (0,0) to (n,0) using steps in {(1,-4), (1,-1), (1,0), (1,1)}.

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 57, 155, 435, 1249, 3645, 10770, 32143, 96747, 293359, 895373, 2748803, 8483035, 26302248, 81896176, 255967640, 802790415, 2525691721, 7968972542, 25209580699, 79942927651, 254077293876, 809192984902, 2582113984084, 8254273128869

Offset: 0

Alois P. Heinz, Oct 06 2021

Alois P. Heinz, Table of n, a(n) for n = 0..1907
Alois P. Heinz, Animation of a(7) = 155 paths
Wikipedia, Counting lattice paths

Cf. A000108, A001006, A025235, A036765, A333069, A333105, A337067.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y-j), j=[-4, -1, 0, 1])))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y - j], {j, {-4, -1, 0, 1}}]]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 3.3640233336410979391691803264403704977... is the root of the equation 256*d^5 - 1280*d^4 + 960*d^3 + 2267*d^2 - 1324*d - 4112 = 0 and c = 0.710307351107763693658610320440791667652705027171696102847138... - Vaclav Kotesovec, Oct 24 2021

cp's OEIS Frontend

A071356 Expansion of (1 - 2*x - sqrt(1 - 4*x - 4*x^2))/(4*x^2).

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A068764 Generalized Catalan numbers 2*x*A(x)^2 -A(x) +1 -x =0.

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A217275 Expansion of 2/(1-x+sqrt(1-2*x-27*x^2)).

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A014431 a(1) = 1, a(2) = 2, a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.

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A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).

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A359364 Triangle read by rows. The Motzkin triangle, the coefficients of the Motzkin polynomials. M(n, k) = binomial(n, k) * CatalanNumber(k/2) if k is even, otherwise 0.

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

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A071356 Expansion of (1 - 2x - sqrt(1 - 4x - 4x^2))/(4x^2).

A068764 Generalized Catalan numbers 2xA(x)^2 -A(x) +1 -x =0.

A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

A014431 a(1) = 1, a(2) = 2, a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-2)*a(2) for n >= 3.

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).

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

A294642 a(n) = n! * [x^n] exp(nx)BesselI(1,2sqrt(2)x)/(sqrt(2)*x).