A127902 -id:A127902 - cp's OEIS frontend

A364522 G.f. satisfies A(x) = 1 + xA(x) + x^5A(x)^5.

1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 258, 518, 1123, 2718, 7008, 18054, 44969, 108189, 255919, 609179, 1482210, 3689155, 9294440, 23419705, 58639835, 145948111, 362721386, 904673836, 2270287636, 5729191861, 14502873988, 36735974548, 93001413353, 235372519273

Offset: 0

Seiichi Manyama, Jul 27 2023

nonn

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 5. - Emanuele Munarini, Jul 11 2024

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

Cf. A001006, A071879, A127902, A364523.

Cf. A182454, A349311.

a(n) = sum(k=0, n\5, binomial(n, 5*k)*binomial(5*k, k)/(4*k+1));

a(n) = Sum_{k=0..floor(n/5)} binomial(n,5*k) * binomial(5*k,k) / (4*k+1).

A364523 G.f. satisfies A(x) = 1 + xA(x) + x^6A(x)^6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 931, 1795, 3550, 7736, 18929, 49505, 130000, 330430, 804271, 1885675, 4327555, 9929515, 23224435, 55907251, 138016906, 345107296, 862546231, 2136402451, 5231163232, 12697101118, 30723857209, 74569942745

Offset: 0

Seiichi Manyama, Jul 27 2023

nonn

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 6. - Emanuele Munarini, Jul 11 2024

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

Cf. A001006, A071879, A127902, A364522.

Cf. A349312, A364472.

a(n) = sum(k=0, n\6, binomial(n, 6*k)*binomial(6*k, k)/(5*k+1));

a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k) * binomial(6*k,k) / (5*k+1).

A369126 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4+x^4) ).

Original entry on oeis.org

1, 4, 22, 140, 970, 7104, 54096, 424008, 3398224, 27721024, 229410328, 1921308272, 16253502512, 138683973120, 1192142838656, 10314377770720, 89749921081280, 784913791336192, 6895599255571840, 60825440855493376, 538507243041624864, 4783482648574893056

Offset: 0

Seiichi Manyama, Jan 13 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A071356, A192132, A369128.

Cf. A127902.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^4+x^4))/x)

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(n+1,k) * binomial(4*n-4*k+4,n-4*k).

D-finite with recurrence -3*(3*n+2)*(3*n+4)*(1746*n-6043)*(n+1)*a(n) +4*(519282*n^4 -1632448*n^3 +319539*n^2 +77803*n-72516)*a(n-1) +16*(-1055610*n^4 +5245655*n^3 -8423433*n^2 +5306215*n-1129842)*a(n-2) +96*(n-2) *(150552*n^3 -673240*n^2 +868987*n -301954)*a(n-3) -64*(n-2) *(n-3) *(174726*n^2 -528221*n +220460)*a(n-4) -512*(7353*n-3733)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 24 2024

A240904 Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3).

Original entry on oeis.org

1, 1, 2, 4, 12, 34, 118, 396, 1508, 5670, 22773, 91337, 381157, 1597683, 6855957, 29599117, 129748149, 572349631, 2551033858, 11435935450, 51651644306, 234472103672, 1070461943299, 4908349870799, 22607570625188, 104515879798724, 484955119433493

Offset: 0

Alois P. Heinz, Apr 14 2014

nonn

			a(0) = 1: the empty path.
a(1) = 1: H.
a(2) = 2: HH, ud.
a(3) = 4: HHH, udH, Hud, uHd.
a(4) = 12: HHHH, udHH, HudH, uHdH, HHud, udud, HuHd, uHHd, uudd, HHUD, udUD, uuuD.
a(5) = 34: HHHHH, udHHH, HudHH, uHdHH, HHudH, ududH, HuHdH, uHHdH, uuddH, HHUDH, udUDH, uuuDH, HHHud, udHud, Hudud, uHdud, HHuHd, uduHd, HuHHd, uHHHd, uudHd, Huudd, uHudd, uuHdd, HHHUD, udHUD, HudUD, uHdUD, HuuuD, uHuuD, uuHuD, HHUHD, udUHD, uuuHD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001006 (without steps U, D), A127902 (without steps U, d), A247748.

Row sums of A247749.

b:= proc(x, y) option remember; `if`(y<0 or x b(n, 0):
seq(a(n), n=0..30);

b[x_, y_] := b[x, y] = If[y<0 || xJean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ c * 5^n / n^(3/2), where c = 0.027245791342943908483296328462... . - Vaclav Kotesovec, Aug 28 2014

A364552 G.f. satisfies A(x) = 1 + xA(x) + x^4A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 21, 39, 78, 169, 373, 808, 1727, 3719, 8153, 18100, 40315, 89770, 200250, 448755, 1010685, 2284295, 5173961, 11740697, 26699780, 60863291, 139045991, 318247190, 729572315, 1675085099, 3851795549, 8869990949, 20453679944, 47223844863

Offset: 0

Seiichi Manyama, Jul 28 2023

Cf. A000108, A071879, A346626.

Cf. A023426, A063019, A127902.

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

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

A365733 G.f. satisfies A(x) = 1 + xA(x)(1 + x^5*A(x)^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 127, 215, 367, 676, 1376, 2982, 6514, 13855, 28407, 56543, 111127, 219918, 444450, 919744, 1933732, 4082467, 8576027, 17861347, 36938427, 76207797, 157652981, 328119005, 687377565, 1446665765, 3050094661, 6427116181

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

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

Cf. A127902, A186996, A215340, A365079.

Cf. A002293, A365735.

a(n) = sum(k=0, n\6, binomial(n-5*k, k)*binomial(n-2*k+1, n-5*k)/(n-2*k+1));

a(n) = Sum_{k=0..floor(n/6)} binomial(n-5*k,k) * binomial(n-2*k+1,n-5*k) / (n-2*k+1) = Sum_{k=0..floor(n/6)} binomial(n-2*k,4*k) * binomial(4*k,k) / (3*k+1).

A369158 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^4) ).

Original entry on oeis.org

1, 2, 5, 14, 43, 142, 495, 1794, 6686, 25436, 98311, 384826, 1522283, 6075838, 24437937, 98956270, 403080170, 1650502292, 6790018182, 28050896964, 116322826479, 484029536374, 2020386475025, 8457397801150, 35495812337114, 149336478356692, 629685490668799

Offset: 0

Seiichi Manyama, Jan 15 2024

nonn

Index entries for reversions of series

Cf. A127902, A369126, A369159.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x^4))/x)

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(n+1,k) * binomial(2*n-2*k+2,n-4*k).

A366646 G.f. A(x) satisfies A(x) = 1 + (x * A(x) / (1 - x))^4.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 10, 20, 39, 88, 228, 600, 1507, 3652, 8866, 22100, 56365, 144656, 369784, 942480, 2408934, 6196280, 16026652, 41571640, 107959654, 280708560, 731349400, 1910098320, 4999759830, 13109582376, 34421585844, 90500370760, 238272324682

Offset: 0

Seiichi Manyama, Oct 15 2023

nonn

Partial sums give A127902.
Cf. A213336, A364410, A366645.
Cf. A002026, A361932.

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

a(n) = Sum_{k=0..floor(n/4)} binomial(n-1,n-4*k) * binomial(4*k,k) / (3*k+1).

A367115 G.f. satisfies A(x) = 1 + 2xA(x) + 2x^4A(x)^4.

Original entry on oeis.org

1, 2, 4, 8, 18, 52, 184, 688, 2512, 8864, 30784, 107648, 384432, 1403872, 5205568, 19443328, 72817856, 273199488, 1027939072, 3883718144, 14741042464, 56189409088, 214931447680, 824443822848, 3169934397184, 12214858010112, 47168251137024

Offset: 0

Seiichi Manyama, Nov 05 2023

Cf. A002293, A127902, A190590, A367114.

Cf. A071356, A367113.

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

a(n) = Sum_{k=0..floor(n/4)} 2^(n-3*k) * binomial(n,4*k) * A002293(k).

A369159 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3+x^4) ).

Original entry on oeis.org

1, 3, 12, 55, 274, 1443, 7905, 44593, 257305, 1511553, 9010170, 54361486, 331336454, 2037132958, 12619056108, 78682008194, 493427982703, 3110202012353, 19693920616872, 125214061831251, 799059649687239, 5116372686471627, 32860439054510610

Offset: 0

Seiichi Manyama, Jan 15 2024

nonn

Index entries for reversions of series

Cf. A127902, A369126, A369158.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3+x^4))/x)

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(n+1,k) * binomial(3*n-3*k+3,n-4*k).

cp's OEIS Frontend

A364522 G.f. satisfies A(x) = 1 + x*A(x) + x^5*A(x)^5.

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A364523 G.f. satisfies A(x) = 1 + x*A(x) + x^6*A(x)^6.

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A369126 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4+x^4) ).

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A240904 Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3).

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A364552 G.f. satisfies A(x) = 1 + x*A(x) + x^4*A(x)^3.

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A365733 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^5*A(x)^3).

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A369158 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^4) ).

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PARI

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A366646 G.f. A(x) satisfies A(x) = 1 + (x * A(x) / (1 - x))^4.

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A367115 G.f. satisfies A(x) = 1 + 2*x*A(x) + 2*x^4*A(x)^4.

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A364522 G.f. satisfies A(x) = 1 + xA(x) + x^5A(x)^5.

A364523 G.f. satisfies A(x) = 1 + xA(x) + x^6A(x)^6.

A364552 G.f. satisfies A(x) = 1 + xA(x) + x^4A(x)^3.

A365733 G.f. satisfies A(x) = 1 + xA(x)(1 + x^5*A(x)^3).

A367115 G.f. satisfies A(x) = 1 + 2xA(x) + 2x^4A(x)^4.