A007863 -id:A007863 - cp's OEIS frontend

A364333 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^6).

1, 2, 17, 216, 3224, 52640, 910452, 16392140, 303996224, 5767278431, 111401778266, 2183535060362, 43319505976084, 868220464851417, 17552981176788200, 357544690982030744, 7330803752675100908, 151172599088871911072, 3133367418601958989295, 65242183918761533467216

Offset: 0

Seiichi Manyama, Jul 18 2023

nonn

Cf. A007863, A069271, A073157, A215654, A215715, A364331.
Cf. A239109, A364339, A364340.
Cf. A200718.

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

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

A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).

Original entry on oeis.org

1, 2, -1, 3, -10, 36, -137, 543, -2219, 9285, -39587, 171369, -751236, 3328218, -14878455, 67030785, -304036170, 1387247580, -6363044315, 29323149825, -135700543190, 630375241380, -2938391049395, 13739779184085, -64430797069375, 302934667061301, -1427763630578197

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Cf. A073157, A007863, A364336, A364337, A364338, A364339, A366326, A366327, A366328.

Cf. A000108, A002212, A112478.

a := proc(n) option remember; if n = 1 then 2 elif n = 2 then -1 else (-3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2))/n fi; end:
seq(a(n), n = 1..30); # Peter Bala, Sep 10 2024

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

G.f.: A(x) = -2*x*(1+x) / (1+x-sqrt((1+x)*(1+5*x))).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n-2,n-k)/(2*k-1).
a(n) ~ -(-1)^n * 5^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 07 2023
From Peter Bala, Sep 10 2024: (Start)
a(n) = 1/(1 - n) * Sum_{k = 0..n} binomial(-n+k, k)*binomial(-n+k+1, n-k) for n not equal to 1. Cf. A007863.
a(n) = Sum_{k = 0..n-2} binomial(-n+k+1, k)*binomial(-n+k+1, n-k)/(-n+k+1) for n >= 2.
P-recursive: n*a(n) = - 3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2) with a(1) = 2 and a(2)  = -1. (End)

A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + xA(x)) (1 + 3xA(x)^2).

Original entry on oeis.org

1, 4, 31, 301, 3274, 38158, 465919, 5883040, 76189177, 1006440238, 13508178448, 183689450959, 2525336086630, 35041483528522, 490125130328455, 6902993856515389, 97814486474787898, 1393470813699724726, 19946461692566594413, 286742046721454817358, 4138001844031453456120

Offset: 0

Paul D. Hanna, Sep 07 2024

nonn

In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then
(C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).
(C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).
(C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).

			G.f. A(x) = 1 + 4*x + 31*x^2 + 301*x^3 + 3274*x^4 + 38158*x^5 + 465919*x^6 + 5883040*x^7 + 76189177*x^8 + 1006440238*x^9 + 13508178448*x^10 + ...
where A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
RELATED SERIES.
Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
B(x) = 1 + 4*x + 15*x^2 + 57*x^3 + 216*x^4 + 819*x^5 + 3105*x^6 + 11772*x^7 + ... + A125145(n)*x^n + ...
where B(x) = (1 + x)/(1 - 3*x - 3*x^2).

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A125145, A215661, A375435, A375436, A375437.

Cf. A007863, A216314, A364374.

{a(n) = my(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + 3*x*(A+x*O(x^n))^2)); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = polcoef( (1/x)*serreverse( x*(1-3*x-3*x^2)/(1+x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2 * 3^j * A^j)*x^m/m))); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies:
(1) A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 3^k * A(x)^k ).
(3) A(x) = (1/x) * Series_Reversion( x*(1 - 3*x - 3*x^2)/(1 + x) ).
(4) A(x) = Sum_{n>=0} A125145(n) * x^n * A(x)^n, where g.f. of A125145 = (1 + x)/(1 - 3*x - 3*x^2).
(5) x = (sqrt(21*A(x)^2 - 6*A(x) + 1) - (1 + 3*A(x)))/(6*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024
a(n) ~ ((36 + (48266 - 714*sqrt(17))^(1/3) + (48266 + 714*sqrt(17))^(1/3))/7)^n / (sqrt(6*Pi*((20517 - 4861*sqrt(17))^(1/3) + (20517 + 4861*sqrt(17))^(1/3) - 42)) * n^(3/2)). - Vaclav Kotesovec, Sep 14 2024

A379021 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^2 ).

Original entry on oeis.org

1, 4, 26, 206, 1813, 17032, 167287, 1697044, 17643322, 186997570, 2012973499, 21948003052, 241883091289, 2690117648372, 30153678822007, 340305271736134, 3863616751855069, 44097785533620550, 505692279260755753, 5823592506326814874, 67320958983831426221

Offset: 0

Seiichi Manyama, Dec 14 2024

nonn

Index entries for reversions of series

Cf. A007863, A379023, A379024.

Cf. A215654, A379022.

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

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379022(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x*A(x)^(3/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A215654.
a(n) = (1/(n+1)) * [x^n] ( (1 + x)/(1 - x - x^2) )^(2*(n+1)).
a(n) = 2 * Sum_{k=0..n} binomial(2*n+k+2,k) * binomial(2*n+k+2,n-k)/(2*n+k+2) = (1/(n+1)) * Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(2*n+k+2,n-k).

A379024 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^4 ).

Original entry on oeis.org

1, 8, 100, 1500, 24846, 438064, 8062518, 153117320, 2978260865, 59031215508, 1187987779084, 24210092837648, 498606095949315, 10361291534825800, 216982960825089730, 4574651332139656108, 97018731642209493810, 2068350691029593934000, 44301394943232879298360

Offset: 0

Seiichi Manyama, Dec 14 2024

nonn

Index entries for reversions of series

Cf. A007863, A379021, A379023.

Cf. A239108, A379026.

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

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379026(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x*A(x)^(5/4)) )^4.
(3) A(x) = B(x)^4 where B(x) is the g.f. of A239108.
a(n) = (1/(n+1)) * [x^n] ( (1 + x)/(1 - x - x^2) )^(4*(n+1)).
a(n) = 4 * Sum_{k=0..n} binomial(4*n+k+4,k) * binomial(4*n+k+4,n-k)/(4*n+k+4) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(4*n+k+4,n-k).

A379327 G.f. A(x) satisfies A(x) = sqrt( (1 + 2xA(x)^2) * (1 + 2xA(x)) ).

Original entry on oeis.org

1, 2, 6, 22, 88, 372, 1634, 7382, 34078, 160034, 762078, 3671178, 17858476, 87599696, 432804190, 2151867226, 10758455224, 54053627604, 272780539742, 1382047628514, 7027307040920, 35848334763884, 183417043984246, 941007480667474, 4839875674661214, 24950493967407850

Offset: 0

Seiichi Manyama, Dec 21 2024

nonn

Cf. A379326, A379328.

Cf. A007863, A379279.

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

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

A011272 Hybrid binary rooted trees with n nodes whose root is labeled by "a".

Original entry on oeis.org

0, 1, 3, 13, 64, 339, 1885, 10851, 64109, 386510, 2368354, 14706331, 92337618, 585239903, 3739309053, 24059542845, 155756019048, 1013801283133, 6630587014935, 43553555324502

Offset: 0

Jean Pallo (pallo(AT)u-bourgogne.fr)

nonn

Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
J. M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50 (1994) 135-145.
Index entries for sequences related to rooted trees

a(n) = A007863(n) - A011270(n).

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

A364340 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^6).

Original entry on oeis.org

1, 2, 15, 179, 2502, 38262, 619991, 10459410, 181771289, 3231782239, 58505593456, 1074766446526, 19984671314164, 375414901633692, 7113886504446443, 135820770971898805, 2610186429457347486, 50452256583633573513, 980187901557594671335, 19130197594133100828170, 374894511736219913097375

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Cf. A239109, A364333, A364339.

Cf. A007863, A198953, A215623, A215624.

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

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

A369630 Expansion of (1/x) * Series_Reversion( x * (1/(1+x^3) - x) ).

Original entry on oeis.org

1, 1, 2, 6, 20, 70, 255, 960, 3707, 14598, 58395, 236626, 969275, 4007041, 16696822, 70053159, 295691622, 1254772103, 5349978803, 22907982780, 98466168572, 424713570017, 1837717336614, 7974744620620, 34698200181696, 151341512079231, 661590732178716

Offset: 0

Seiichi Manyama, Jan 28 2024

nonn

Index entries for reversions of series

Cf. A007863, A199874, A369631.

Cf. A226974.

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

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

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

cp's OEIS Frontend

A364333 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^6).

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A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).

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A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).

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A379021 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^2 ).

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A379024 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^4 ).

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A379327 G.f. A(x) satisfies A(x) = sqrt( (1 + 2*x*A(x)^2) * (1 + 2*x*A(x)) ).

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A011272 Hybrid binary rooted trees with n nodes whose root is labeled by "a".

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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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A364340 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^6).

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A369630 Expansion of (1/x) * Series_Reversion( x * (1/(1+x^3) - x) ).

A364333 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^6).

A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + xA(x)) (1 + 3xA(x)^2).

A379327 G.f. A(x) satisfies A(x) = sqrt( (1 + 2xA(x)^2) * (1 + 2xA(x)) ).

A364340 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^6).