A007564 -id:A007564 - cp's OEIS frontend

A364923 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - 2x*A(x)^3).

1, 1, 6, 48, 442, 4419, 46626, 511032, 5761650, 66394596, 778518552, 9258850440, 111417705702, 1354135251538, 16598001854700, 204945037918800, 2546849778687138, 31828936270676172, 399777371427582024, 5043824569861127808, 63892650400004356776

Offset: 0

Seiichi Manyama, Aug 12 2023

nonn

Cf. A007564, A364924.

Cf. A243659.

A364923 := proc(n)
    add( 3^k*(-2)^(n-k)*binomial(n,k)*binomial(3*n+k+1,n)/(3*n+k+1),k=0..n) ;
end proc:
seq(A364923(n),n=0..80); # R. J. Mathar, Aug 16 2023

a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1));

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
D-finite with recurrence +270*n*(3*n-1)*(3*n+1)*a(n) +(-9463*n^3 -45948*n^2 +88297*n -35478)*a(n-1) +36*(-9017*n^3 +49691*n^2 -90408*n +54354)*a(n-2) +48*(53*n^3 +1724*n^2 -11161*n +16518)*a(n-3) +576*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - R. J. Mathar, Aug 16 2023

A224071 Number of Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

Original entry on oeis.org

1, 2, 5, 15, 52, 201, 841, 3726, 17213, 82047, 400600, 1993377, 10071777, 51532938, 266462229, 1390174911, 7308741084, 38682855225, 205940368441, 1102091393574, 5925177392573, 31987877317887, 173337754977904

Offset: 0

José Luis Ramírez Ramírez, Mar 30 2013

nonn

Hankel transform is A006215. Invert transform of A155069. - Michael Somos, Apr 02 2013

			a(2) = 5 because we have HH, UDH, HUD, UDUD and UUDD.
G.f. = 1 + 2*x + 5*x^2 + 15*x^3 + 52*x^4 + 201*x^5 + 841*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Bloom and S. Elizalde, Pattern avoidance in matchings and partitions, arXiv:1211.3442 [math.CO], 2012; Theorem 6.1.
Paul Barry, A study of Integer Sequences, Riordan Arrays, Pascal-like Arrays and Hankel Transforms, Ph.D Thesis, University College, Cork, Republic of Ireland, 2009.
Arnauld Mesinga Mwafise, Computational and Combinatorial Enumeration of Poset Matrices, 2024. See p. 8.

Cf. A001003, A007564, A010892, A059231.

CoefficientList[Series[4/(3-5*x+Sqrt[x^2-6*x+1]), {x, 0, 20}], x] (* Vaclav Kotesovec, May 23 2013 *)
a[ n_] := SeriesCoefficient[ (3 - 5 x - Sqrt[ 1 - 6 x + x^2]) / (2 - 6 x + 6 x^2), {x, 0, n}]; (* Michael Somos, Mar 28 2014 *)

a(n):=sum((k+1)*((-1)^floor((k+2)/3)+(-1)^floor((k+1)/3))*sum(binomial(n+1,n-k-i)*binomial(n+i,n),i,0,n-k),k,0,n)/(2*(n+1)); /* Vladimir Kruchinin, Mar 08 2016*/

z='z+O('z^66); Vec(4/(3-5*z+sqrt(1-6*z+z^2))) /* Joerg Arndt, Mar 30 2013 */

G.f.: 4/(3-5*x+sqrt(1-6*x+x^2)).
Recurrence: n*a(n) = 9*(n-1)*a(n-1) - 2*(11*n-15)*a(n-2) + 3*(7*n-12)*a(n-3) - 3*(n-3)*a(n-4). - Vaclav Kotesovec, May 23 2013
a(n) ~ sqrt(884+627*sqrt(2)) * (3+2*sqrt(2))^n / (98*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, May 23 2013
0 = +a(n)*(+9*a(n+1) - 144*a(n+2) + 174*a(n+3) - 81*a(n+4) + 12*a(n+5)) + a(n+1)*(+18*a(n+1) + 399*a(n+2) - 597*a(n+3) + 318*a(n+4) - 57*a(n+5)) + a(n+2)*(-300*a(n+2) + 538*a(n+3) - 255*a(n+4) + 52*a(n+5)) + a(n+3)*(-126*a(n+3) + 73*a(n+4) - 18*a(n+5)) + a(n+4)*(+a(n+5)) if n>=0. - Michael Somos, Mar 28 2014
a(n) = Sum_{k=0..n}((k+1)*((-1)^floor((k+2)/3)+(-1)^floor((k+1)/3))*Sum_{i=0..n-k}(binomial(n+1,n-k-i)*binomial(n+i,n)))/(2*(n+1)). - Vladimir Kruchinin, Mar 08 2016

A336729 G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 + 3 * x * A(x)).

Original entry on oeis.org

1, 1, -2, 1, 10, -38, 28, 289, -1262, 1054, 11044, -51302, 45604, 482068, -2319176, 2140129, 22753378, -111964106, 105927508, 1130780062, -5652760340, 5444054956, 58291068808, -294808277414, 287740874260, 3088109246572, -15758505143192, 15541351662484, 167103084713608

Offset: 0

Seiichi Manyama, Aug 02 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=3 of A336727.

Cf. A007564, A116091.

a[0] = 1; a[n_] := Sum[(-3)^(n - k) * Binomial[n, k] * Binomial[n , k - 1], {k, 1, n}] / n; Array[a, 29, 0] (* Amiram Eldar, Aug 02 2020 *)

{a(n) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+3*x*A)); polcoef(A, n)}

{a(n) = if(n==0, 1, sum(k=1, n, (-3)^(n-k)*binomial(n, k)*binomial(n, k-1))/n)}

N=40; x='x+O('x^N); Vec(2/(1-4*x+sqrt(1+4*x+16*x^2)))

{a(n) = sum(k=0, n, (-3)^k*4^(n-k)*binomial(n, k)*binomial(n+k, n)/(k+1))}

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-3)^(n-k) * binomial(n,k) * binomial(n,k-1) for n > 0.
G.f.: 2/(1 - 4*x + sqrt(1 + 4*x + 16*x^2)).
a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(n,k) * binomial(n+k,n)/(k+1).
(n+1) * a(n) = -2 * (2*n-1) * a(n-1) - 16 * (n-2) * a(n-2) for n>1. - Seiichi Manyama, Aug 08 2020
a(n) ~ 2^(2*n - 1/2) * ((sqrt(3) + 1)*sin(2*Pi*n/3) + (sqrt(3) - 1)*cos(2*Pi*n/3)) / (3^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 04 2020

A371363 Expansion of (1/x) * Series_Reversion( x * (1-3x)^3 / (1-2x) ).

Original entry on oeis.org

1, 7, 85, 1261, 20788, 365845, 6731758, 127938625, 2491921516, 49480794460, 997897366717, 20384025765619, 420869454302620, 8769197604091246, 184151509243984300, 3893585866824069577, 82817275938125471548, 1770880435886367151060

Offset: 0

Seiichi Manyama, Mar 19 2024

nonn

Index entries for reversions of series

Cf. A007564, A371362.

Cf. A369023, A371365.

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-3*x)^3/(1-2*x))/x)

a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+k+2, k)*binomial(3*n+1, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(3*n+k+2,k) * binomial(3*n+1,n-k).

A234939 Coefficients of Hilbert series for suboperad of bicolored noncrossing configurations generated by a triangle with colored base and at least one more colored edge and a triangle with one colored non-base edge.

Original entry on oeis.org

1, 2, 8, 38, 200, 1124, 6608, 40142, 249992, 1587548, 10241264, 66926204, 442120016, 2947660616, 19808372384, 134030802782, 912385334792, 6244056445868, 42935538999728, 296493196682036, 2055313327353200, 14297177397185912, 99769106353379168, 698228176760193068

Offset: 1

N. J. A. Sloane, Jan 04 2014

nonn

Frédéric Chapoton and Samuele Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv preprint arXiv:1310.4521 [math.CO], 2013-2014.

Cf. A234938, A052701, A007863, A006013, A006318, A007564.

a(n) = 2 * A007564(n-1) for n > 1 [from Chapoton & Giraudo, Proposition 3.8]. - Andrey Zabolotskiy, Feb 01 2025

Terms a(9) onwards added and name clarified by Andrey Zabolotskiy, Feb 02 2025

A364924 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - 2x*A(x)^4).

Original entry on oeis.org

1, 1, 7, 67, 743, 8970, 114445, 1517976, 20722023, 289224355, 4108588558, 59207805442, 863439906413, 12718638581368, 188960182480440, 2828238875318256, 42605850936335463, 645497106959662857, 9829072480785776101, 150345303724987825021

Offset: 0

Seiichi Manyama, Aug 12 2023

nonn

Cf. A007564, A364923.

Cf. A243667.

a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1));

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(4*n+k+1,n) / (4*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n,k) * binomial(4*n,k-1) for n > 0.

A371362 Expansion of (1/x) * Series_Reversion( x * (1-3x)^2 / (1-2x) ).

Original entry on oeis.org

1, 4, 31, 298, 3199, 36742, 441748, 5489554, 69945295, 908836768, 11996580199, 160418984656, 2168512922692, 29584600414168, 406823494817560, 5632906243123090, 78465351036084655, 1098851032467132484, 15461857967408794333, 218490450548650811914

Offset: 0

Seiichi Manyama, Mar 19 2024

nonn

Index entries for reversions of series

Cf. A007564, A371363.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-3*x)^2/(1-2*x))/x)

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

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(2*n+k+1,k) * binomial(2*n,n-k).

A243693 Number of Hyposylvester classes of 3-multiparking functions of length n.

Original entry on oeis.org

1, 1, 5, 32, 233, 1833, 15180, 130392, 1151057, 10378883, 95182445, 885053524, 8324942620, 79071217228, 757310811912, 7305728683824, 70923966744609, 692370887676567, 6792525607165935, 66933512163735000, 662190712902022017, 6574831459429388169, 65494637699437417584

Offset: 0

N. J. A. Sloane, Jun 14 2014

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020. See Fig. 27.
Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.1.

Cf. A243694, A243695.

Cf. A007564, A003168, A364923, A364924.

a := proc(n) option remember; if n <= 1 then return 1 fi;
(a(n - 2)*(-800*n^3 + 3024*n^2 - 3184*n + 672) + a(n - 1)*(3275*n^3 - 7467*n^2 +
5038*n - 1008))/(300*n^3 - 234*n^2 - 192*n) end:
seq(a(n), n = 0..22);  # Peter Luschny, Apr 13 2024

a[n_] := 3^(n - Boole[n>0]) Hypergeometric2F1[1 - n, -2 n, 2, 1/3];
Table[a[n], {n, 0, 22}]  (* Peter Luschny, Apr 12 2024 *)

a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1)); \\ Seiichi Manyama, Aug 12 2023

From Seiichi Manyama, Aug 12 2023: (Start)
The following statements are equivalent:
The g.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - 2*x*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n, k) * binomial(2*n+k+1, n) / (2*n + k + 1).
a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n, k) * binomial(2*n, k-1) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n, k)*binomial(3*n-k, n-1-k) for n > 0.
 (End)
The above formula is proved in Theorem 4.1 of the Jun Yan link to be the number of Hyposylvester classes of 3-multiparking functions of length n. - Jun Yan, Apr 12 2024
a(n) ~ 2^(5*n+1) / (sqrt(5*Pi) * n^(3/2) * 3^(n+1)). - Vaclav Kotesovec, Apr 12 2024
a(n) = 3^(n - 1) * hypergeom([1 - n, -2*n], [2], 1/3) for n > 0. - Peter Luschny, Apr 12 2024
G.f. A(x) = 1 + series_reversion( x/((1 + 3*x)*(1 + x)^2) ). - Peter Bala, Sep 10 2024

Name clarified by Jun Yan, Apr 12 2024

A250307 Number of 2-colored Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

Original entry on oeis.org

1, 3, 10, 37, 152, 690, 3422, 18257, 103144, 608730, 3713524, 23235490, 148281656, 961255200, 6311395814, 41878914665, 280365966232, 1891270498050, 12842102343820, 87703053156406, 601999871121200, 4150859861430252, 28736613316786220, 199671324115916570

Offset: 0

José Luis Ramírez Ramírez, Apr 20 2015

nonn

			a(2) = 10 because we have H1H1, H1H2, H2H1, H2H2, UDH1, UDH2, H1UD, H2UD, UDUD and UUDD.

Cf. A007564, A224071.

CoefficientList[Series[1/(1 - 2 x - x (1 + 2 x - Sqrt[1 - 8 x + 4 x^2])/(6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 21 2015 *)

a(n):=sum((k+1)*(sum(2^j*(-1)^(-k+j)*binomial(k-j,j),j,0,k))*sum(binomial(j,-n-k+2*j-2)*4^(-n-k+2*j-2)*3^(n-j+1)*binomial(n+1,j),j,0,n+1),k,0,n)/(n+1); /* Vladimir Kruchinin, Mar 08 2016 */

G.f.: 6/(5-14*x+sqrt(1-8*x+4*x^2))=1/(1-2*x-x*F(x)), where F(x) is the g.f. of the sequence A007564.
a(n) ~ sqrt(1275+746*sqrt(3)) * (2*(2+sqrt(3)))^n / (121*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
a(n) = Sum_{k=0..n}((k+1)*(Sum_{j=0..k}(2^j*(-1)^(-k+j)*binomial(k-j,j)))*Sum_{j=0..n+1}(binomial(j,-n-k+2*j-2)*4^(-n-k+2*j-2)*3^(n-j+1)*binomial(n+1,j)))/(n+1). - Vladimir Kruchinin, Mar 08 2016
Conjecture: 2*n*a(n) +3*(-9*n+8)*a(n-1) +4*(28*n-39)*a(n-2) +4*(-43*n+81)*a(n-3) +64*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 24 2016

More terms from Vincenzo Librandi, Apr 21 2015

cp's OEIS Frontend

A364923 G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 - 2*x*A(x)^3).

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A224071 Number of Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

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A336729 G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 + 3 * x * A(x)).

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A371363 Expansion of (1/x) * Series_Reversion( x * (1-3*x)^3 / (1-2*x) ).

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A234939 Coefficients of Hilbert series for suboperad of bicolored noncrossing configurations generated by a triangle with colored base and at least one more colored edge and a triangle with one colored non-base edge.

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A364924 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - 2*x*A(x)^4).

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A371362 Expansion of (1/x) * Series_Reversion( x * (1-3*x)^2 / (1-2*x) ).

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A243693 Number of Hyposylvester classes of 3-multiparking functions of length n.

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A364923 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - 2x*A(x)^3).

A371363 Expansion of (1/x) * Series_Reversion( x * (1-3x)^3 / (1-2x) ).

A364924 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - 2x*A(x)^4).

A371362 Expansion of (1/x) * Series_Reversion( x * (1-3x)^2 / (1-2x) ).