A007788 -id:A007788 - cp's OEIS frontend

A007863 Number of hybrid binary trees with n internal nodes.

1, 2, 7, 31, 154, 820, 4575, 26398, 156233, 943174, 5785416, 35955297, 225914342, 1432705496, 9158708775, 58954911423, 381806076426, 2485972170888, 16263884777805, 106858957537838, 704810376478024, 4664987368511948, 30974829705533240, 206266525653071416

Offset: 0

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

nonn

From Benoit Jubin, May 27 2012: (Start)
Definition of hybrid binary trees:
An (a,n)-labeled binary tree is a binary tree where each internal node is labeled by "a" (for associative) or "n" (for nonassociative). We define on the set of (a,n)-labeled binary trees with a given number of nodes an equivalence relation as follows: denote a tree with a root labeled "a" with left subtree A and right subtree B by AaB. Then we declare the trees (AaB)aC and Aa(BaC) equivalent, and two trees are equivalent if and only if one can go from one to the other by doing such transformations within any of their subtrees.
A hybrid binary tree is an equivalence class of (a,n)-labeled binary trees under this relation. (End)
Also the number of Dyck n-paths with up steps colored in two ways (N or A) and avoiding AA. The 7 Dyck 2-paths are NDND, NDAD, ADND, ADAD, NNDD, NADD, and ANDD. - David Scambler, May 21 2012

			G.f. = 1 + 2*x + 7*x^2 + 31*x^3 + 154*x^4 + 820*x^5 + 4575*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
F. Chapoton, S. Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv preprint arXiv:1310.4521 [math.CO], 2013-2014.
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Preprint submitted to Ann. Sci. Math. Quebec, 1994. (Annotated scanned copy)
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235.
J. M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50, 1994, 135-145.
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)
Index entries for sequences related to rooted trees

Cf. A007788, A011365.

Column k=2 of A245049.

taylor_solve_choose_order:true; taylor_solve( A^3*X^2+A^2*X+A*(X-1)+1,A,X,0,[ 20 ]);

A:= proc(n) option remember; if n=0 then 1 else convert(series((x^2 *A(n-1)^3 +x*A(n-1)^2 +1)/ (1-x), x=0,n+1), polynom) fi end: a:= n-> coeff(A(n), x, n): seq(a(n), n=0..30); # Alois P. Heinz, Aug 22 2008

InverseSeries[Series[(y-y^2-y^3)/(1+y), {y, 0, 24}], x] (* then A(x)=y(x)/x . - Len Smiley, Apr 14 2000 *)
Table[ HypergeometricPFQ[{-n, 1 + n, 2 + n}, {1, 3/2}, -(1/4)], {n,0,23}] (* Olivier Gérard, Apr 23 2009 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[{-n, 1 + n, 2 + n}, {1, 3/2}, -1/4]]; (* Michael Somos, Dec 31 2014 *)

{a(n) = if( n<0, 0, sum(k=0, n, binomial(n+k, n) * binomial(n+k+1, n-k)) / (n+1))};

{a(n) = local(A = 1 + x + x * O(x^n)); for(i=1, n, A = 1 + x * (A + A^2) + x^2 * A^3); polcoeff(A, n)};

{a(n) = local(A=1+x); for(i=1, n, A = exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2 * (A + x * O(x^n))^j) * x^m / m))); polcoeff(A, n, x)};

G.f. A(x) satisfies: x^2*A(x)^3 + x*A(x)^2 + (-1+x)*A(x) + 1 = 0.
a(n) = 3F2({-n, n+1, n+2 } ; {1, 3/2})( -(1/4) ). - Olivier Gérard, Apr 23 2009
a(n) = (1/(n+1))*Sum_{k=0..n} binomial(n+k,n)*binomial(n+k+1,n-k). - Vladimir Kruchinin, Dec 24 2010
G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*A(x)^k] * x^n/n ). - Paul D. Hanna, Feb 13 2011
The formal inverse of the g.f. A(x) is (sqrt(5*x^2 - 2*x + 1) - (1+x))/(2*x^2). - Paul D. Hanna, Aug 21 2012
The radius of convergence of g.f. A(x) is r = 0.1407810125... with A(r) = 2.1107712092... such that y=A(r) satisfies 5*y^3 - 12*y^2 + 4*y - 2 = 0. - Paul D. Hanna, Aug 21 2012
D-finite with recurrence: 45*n*(n+1)*a(n) - 2*n*(157*n-71)*a(n-1) + 12*(-3*n^2+15*n-14)*a(n-2) + 2*(-14*n^2+43*n-21)*a(n-3) - 4*(n-3)*(2*n-7)*a(n-4) = 0. - R. J. Mathar, Jun 03 2014
Recurrence (of order 3): 5*n*(n+1)*(35*n-62)*a(n) = 6*n*(210*n^2 - 477*n + 181)*a(n-1) - 4*n*(35*n^2 - 132*n + 115)*a(n-2) + 2*(n-2)*(2*n-5)*(35*n-27)*a(n-3). - Vaclav Kotesovec, Jul 11 2014
a(n) ~ sqrt((s*(1+s+2*r*s^2))/(1+3*r*s)) / (2*sqrt(Pi) * r^n * n^(3/2)), where r = 52/(3*(181 + 105*sqrt(105))^(1/3)) - 1/6*(181 + 105*sqrt(105))^(1/3) + 1/3 = 0.1407810125885522212..., s = 1/15*(12 + (1323 - 105*sqrt(105))^(1/3) + (21*(63 + 5*sqrt(105)))^(1/3)) = 2.110771209224758867... . - Vaclav Kotesovec, Jul 11 2014

A144015 Expansion of e.g.f. 1/(1 - sin(4*x))^(1/4).

Original entry on oeis.org

1, 1, 5, 29, 265, 3001, 42125, 696149, 13296145, 287706481, 6959431445, 186061833869, 5448382252825, 173418192216361, 5961442393047965, 220112963745653189, 8687730877758518305, 365023930617143804641, 16266420334783460443685, 766297734521812843642109

Offset: 0

Paul D. Hanna, Sep 09 2008

nonn

Row sums of A186492 - Peter Bala, Feb 22 2011.

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 29*x^3/3! + 265*x^4/4! + 3001*x^5/5! +...
log(A(x)) = x + 4*x^2/2! + 16*x^3/3! + 128*x^4/4! + 1280*x^5/5! +...
A(x)^2/A(-x)^2 = 1 + 4*x + 16*x^2/2! + 128*x^3/3! +...+ 4^n*A000111(n)*x^n/n! +...
O.g.f.: 1/(1-x - 4*1*1*x^2/(1-5*x - 4*2*3*x^2/(1-9*x - 4*3*5*x^2/(1-13*x - 4*4*7*x^2/(1-17*x - 4*5*9*x^2/(1-...)))))) [continued fraction by Sergei Gladkovskii].

Cf. A000111, A001586, A007788, A186492, A230134, A227544, A230114, A235329.

Cf. A007696, A136630.

CoefficientList[Series[1/(1-Sin[4*x])^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(2*X)-sin(2*X))^(-1/2), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=exp(intformal(A^2/subst(A^2,x,-x))));n!*polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))

/* From A'(x) = A(x)^3 / A(-x)^2: */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^3/subst(A, x, -x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

/* 1/sqrt(1-2*Series_Reversion(Integral 1/sqrt(1+4*x-4*x^2) dx)): */
{a(n)=local(A=1);A=1/sqrt(1-2*serreverse(intformal(1/sqrt(1+4*x-4*x^2 +x*O(x^n)))));n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007696(n) = prod(k=0, n-1, 4*k+1);
a(n) = sum(k=0, n, a007696(k)*(4*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies:
(1) A(x) = (cos(2*x) - sin(2*x))^(-1/2).
(2) A(x)^2/A(-x)^2 = 1/cos(4*x) + tan(4*x).
(3) A(x) = exp( Integral A(x)^2/A(-x)^2 dx).
(4) A'(x) = A(x)^3/A(-x)^2 with A(0) = 1.
(5) A(x) = 1/sqrt(1 - 2*Series_Reversion( Integral 1/sqrt(1+4*x-4*x^2) dx )).
G.f.: 1/G(0) where G(k) = 1 - x*(4*k+1) - 4*x^2*(k+1)*(2*k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013.
a(n) ~ 2^(3*n+5/4)*n^n/(exp(n)*Pi^(n+1/2)). - Vaclav Kotesovec, Jun 26 2013
a(n) = Sum_{k=0..n} A007696(k) * (4*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A227544 Expansion of e.g.f. 1/(1 - sin(6*x))^(1/6).

Original entry on oeis.org

1, 1, 7, 55, 721, 11761, 240247, 5801095, 162512161, 5171130721, 184337942887, 7275081518935, 314918762166001, 14834964193292881, 755507853144691927, 41362173671901329575, 2422478811455080626241, 151132171549872325122241, 10006051653759338150151367, 700695219796759105368529015

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)). - Vaclav Kotesovec, Jan 03 2014

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 55*x^3/3! + 721*x^4/4! + 11761*x^5/5! + ...
where A(x)^3 = 1 + 3*x + 27*x^2/2! + 297*x^3/3! + 4617*x^4/4! + 87723*x^5/5! + ...
and 1/A(x)^3 = 1 - 3*x - 9*x^2/2! + 27*x^3/3! + 81*x^4/4! - 243*x^5/5! + ...
which illustrates 1/A(x)^3 = cos(3*x) - sin(3*x).
O.g.f.: 1/(1-x - 6*1*1*x^2/(1-7*x - 6*2*4*x^2/(1-13*x - 6*3*7*x^2/(1-19*x - 6*4*10*x^2/(1-25*x - 6*5*13*x^2/(1-...)))))), a continued fraction.

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A235128 (p=7), A230114 (p=8).

Cf. A008542, A136630.

CoefficientList[Series[1/(1-Sin[6*x])^(1/6), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(3*X)-sin(3*X))^(-1/3), n)}
for(n=0,20,print1(a(n),", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^3/subst(A^3, x, -x)))); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a008542(n) = prod(k=0, n-1, 6*k+1);
a(n) = sum(k=0, n, a008542(k)*(6*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies:
(1) A(x) = (cos(3*x) - sin(3*x))^(-1/3).
(2) A(x)^3/A(-x)^3 = 1/cos(6*x) + tan(6*x).
(3) A(x) = exp( Integral A(x)^3/A(-x)^3 dx ).
O.g.f.: 1/G(0) where G(k) = 1 - (6*k+1)*x - 6*(k+1)*(3*k+1)*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * 2^(2*n+1/2) * 3^n / (Gamma(1/3) * n^(2/3) * Pi^(n+1/3)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A008542(k) * (6*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A230114 Expansion of e.g.f. 1/(1 - sin(8*x))^(1/8).

Original entry on oeis.org

1, 1, 9, 89, 1521, 32401, 869049, 27608489, 1019581281, 42824944801, 2017329504489, 105299243488889, 6032850630082641, 376363074361201201, 25396689469918450329, 1843101478742259481289, 143145930384321475601601, 11846611289341729822881601, 1040750126963789832859930569

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) is a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)). - Vaclav Kotesovec, Jan 03 2014

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 89*x^3/3! + 1521*x^4/4! + 32401*x^5/5! + ...
where A(x)^4 = 1 + 4*x + 48*x^2/2! + 704*x^3/3! + 14592*x^4/4! + 369664*x^5/5! + ...
and 1/A(x)^4 = 1 - 4*x - 16*x^2/2! + 64*x^3/3! + 256*x^4/4! - 1024*x^5/5! + ...
which illustrates 1/A(x)^4 = cos(4*x) - sin(4*x).
O.g.f.: 1/(1-x - 8*1*1*x^2/(1-9*x - 8*2*5*x^2/(1-17*x - 8*3*9*x^2/(1-25*x - 8*4*13*x^2/(1-33*x - 8*5*17*x^2/(1-...)))))), a continued fraction.

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A235128 (p=7).

Cf. A045755, A136630.

CoefficientList[Series[1/(1-Sin[8*x])^(1/8), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(4*X)-sin(4*X))^(-1/4), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^4/subst(A^4, x, -x)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a045755(n) = prod(k=0, n-1, 8*k+1);
a(n) = sum(k=0, n, a045755(k)*(8*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies:
(1) A(x) = (cos(4*x) - sin(4*x))^(-1/4).
(2) A(x)^4/A(-x)^4 = 1/cos(8*x) + tan(8*x).
(3) A(x) = exp( Integral A(x)^4/A(-x)^4 dx ).
O.g.f.: 1/G(0) where G(k) = 1 - (8*k+1)*x - 8*(k+1)*(4*k+1)*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * 2^(4*n+3/8) / (Gamma(1/4) * n^(3/4) * Pi^(n+1/4)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A045755(k) * (8*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A230134 Expansion of e.g.f. 1/(1 - sin(5*x))^(1/5).

Original entry on oeis.org

1, 1, 6, 41, 456, 6301, 108576, 2207981, 52012416, 1390239481, 41593598976, 1376769180401, 49955931795456, 1971671764875541, 84095262825824256, 3854514200269774901, 188942180401957502976, 9863099585213327293681, 546266997049408050364416, 31993839349571172423492281

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 41*x^3/3! + 456*x^4/4! + 6301*x^5/5! +...
O.g.f.: 1/(1-x - 5*1*2/2*x^2/(1-6*x - 5*2*7/2*x^2/(1-11*x - 5*3*12/2*x^2/(1-16*x - 5*4*17/2*x^2/(1-21*x - 5*5*22/2*x^2/(1-...)))))), a continued fraction.

Cf. A001586, A007788, A144015, A227544, A230114.

Cf. A008548, A136630.

CoefficientList[Series[1/(1-Sin[5*x])^(1/5), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((1-sin(5*X))^(-1/5), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^(5/2)/subst(A^(5/2), x, -x)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a008548(n) = prod(k=0, n-1, 5*k+1);
a(n) = sum(k=0, n, a008548(k)*(5*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies: A(x) = (cos(5*x/2) - sin(5*x/2))^(-2/5).
O.g.f.: 1/G(0) where G(k) = 1 - (5*k+1)*x - 5*(k+1)*(5*k+2)/2*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * sqrt(5+sqrt(5)) * Gamma(3/5) * 2^(n-9/10) * 5^n / (n^(3/5) * Pi^(n+7/5)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A008548(k) * (5*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A235128 Expansion of e.g.f. 1/(1 - sin(7*x))^(1/7).

Original entry on oeis.org

1, 1, 8, 71, 1072, 20161, 476288, 13315751, 432387712, 15959926081, 660372282368, 30265936565831, 1522069164439552, 83327826089289601, 4933286107483701248, 314052936209639958311, 21392225375507849838592, 1552501782546292090638721, 119588747474281844162428928

Offset: 0

Vaclav Kotesovec, Jan 03 2014

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)).

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A230114 (p=8).

Cf. A045754, A136630.

CoefficientList[Series[1/(1-Sin[7*x])^(1/7), {x, 0, 20}], x] * Range[0, 20]!

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a045754(n) = prod(k=0, n-1, 7*k+1);
a(n) = sum(k=0, n, a045754(k)*(7*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

a(n) ~ n! * 2^(n+3/7) * 7^n / (Gamma(2/7) * n^(5/7) * Pi^(n+2/7)).

a(n) = Sum_{k=0..n} A045754(k) * (7*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A235135 Expansion of e.g.f. 1/(1 - sinh(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 37, 424, 6241, 113824, 2460277, 61504384, 1746727201, 55545439744, 1955176596517, 75470959673344, 3169939381277761, 143927870364811264, 7024566555751464757, 366742587098140770304, 20394984041632355113921, 1203587891190987380752384, 75124090160952970927512997

Offset: 0

Vaclav Kotesovec, Jan 03 2014

Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).

Cf. A006154, A235134.

Cf. A007559, A007788, A136630, A235132.

CoefficientList[Series[1/(1-Sinh[3*x])^(1/3), {x, 0, 20}], x] * Range[0, 20]!

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*3^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

a(n) ~ n! * 3^n / (Gamma(1/3) * 2^(1/6) * n^(2/3) * (log(1+sqrt(2)))^(n+1/3)).

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025

A385307 Expansion of e.g.f. 1/(1 - 3 * sin(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 27, 264, 3361, 52704, 981707, 21176704, 519150241, 14255163904, 433384277787, 14451212550144, 524406240059521, 20572970822959104, 867641565719168267, 39145118179183427584, 1881294510800399083201, 95950279080398196834304, 5176039012712211526485147

Offset: 0

Seiichi Manyama, Jun 24 2025

nonn

Cf. A000111, A385306.
Cf. A380017, A385305, A385309, A385311.
Cf. A007559, A007788, A136630.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*I^(n-k)*a136630(n, k));

a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

a(n) ~ n! / (sqrt(2) * Gamma(1/3) * n^(2/3) * arcsin(1/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 28 2025

A385422 Expansion of e.g.f. 1/(1 - arcsin(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 37, 424, 6889, 129376, 3004597, 78196864, 2363157937, 78520720384, 2924352594373, 118146438461440, 5232528466643737, 248845526415892480, 12778931460471237397, 699044652076991610880, 40846771050451091426785, 2526020027235443981025280

Offset: 0

Seiichi Manyama, Jun 28 2025

nonn

Cf. A189780, A385421.

Cf. A007559, A007788, A235135, A385343, A385420.

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

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A385343(n,k).

a(n) ~ sqrt(2*Pi) * cos(1)^(1/3) * 3^n * n^(n - 1/6) / (Gamma(1/3) * sin(1)^(n + 1/3) * exp(n)). - Vaclav Kotesovec, Jun 28 2025

A385896 Array read by ascending antidiagonals: A(n, k) = k! * [x^k] (1 - sin(n*x))^(-1/n) for n > 0, A(0, k) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 11, 16, 1, 1, 1, 5, 19, 57, 61, 1, 1, 1, 6, 29, 136, 361, 272, 1, 1, 1, 7, 41, 265, 1201, 2763, 1385, 1, 1, 1, 8, 55, 456, 3001, 13024, 24611, 7936, 1, 1, 1, 9, 71, 721, 6301, 42125, 165619, 250737, 50521, 1

Offset: 0

Peter Luschny, Jul 20 2025

			Table starts:
  [0] 1, 1, 1,  1,    1,     1,      1, ... [A000012]
  [1] 1, 1, 2,  5,   16,    61,    272, ... [A000111]
  [2] 1, 1, 3, 11,   57,   361,   2763, ... [A001586]
  [3] 1, 1, 4, 19,  136,  1201,  13024, ... [A007788]
  [4] 1, 1, 5, 29,  265,  3001,  42125, ... [A144015]
  [5] 1, 1, 6, 41,  456,  6301, 108576, ... [A230134]
  [6] 1, 1, 7, 55,  721, 11761, 240247, ... [A227544]
  [7] 1, 1, 8, 71, 1072, 20161, 476288, ... [A235128]
  [8] 1, 1, 9, 89, 1521, 32401, 869049, ... [A230114]
     [A000027]  | [A187277] | [A385898].
            [A028387]   [A385897]
.
Seen as a triangle:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1, 1;
  [3] 1, 1, 2,  1;
  [4] 1, 1, 3,  5,   1;
  [5] 1, 1, 4, 11,  16,    1;
  [6] 1, 1, 5, 19,  57,   61,    1;
  [7] 1, 1, 6, 29, 136,  361,  272,    1;
  [8] 1, 1, 7, 41, 265, 1201, 2763, 1385, 1;

Rows: A000012, A000111, A001586, A007788, A144015, A230134, A227544, A235128, A230114.
Columns: A000027, A028387, A187277, A385897, A385898.
Cf. A135278.

MAX := 16: ser := n -> series((1 - sin(n*x))^(-1/n), x, MAX):
A := (n, k) -> if n = 0 then 1 else k!*coeff(ser(n), x, k) fi:
seq(lprint(seq(A(n, k), k = 0..8)), n = 0..8);

T[n_, k_, m_] := T[n, k, m] =
  Which[
    n <  0 || k <  0, 0,
    n == 0 && k == 0, 1,
    k == 0, T[n - 1, n - 1, m],
    True, T[n, k - 1, m] + m*T[n - 1, n - k - 1, m]
];
A[n_, k_] := T[k, k, n - k];
Table[A[n, k], {n, 0, 10}, {k, 0, n}] // Flatten

A(n, k) = T(k, k, n - k) where T(n, k, m) = T(n, k-1, m) + m * T(n-1, n-k-1, m) for k > 0, T(n, 0, m) = T(n-1, n-1, m), and T(0, 0, m) = 1.

Column n is a linear recurrence with kernel [(-1)^k*A135278(n, k), k = 0..n].

cp's OEIS Frontend

A007863 Number of hybrid binary trees with n internal nodes.

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A144015 Expansion of e.g.f. 1/(1 - sin(4*x))^(1/4).

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A227544 Expansion of e.g.f. 1/(1 - sin(6*x))^(1/6).

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A230114 Expansion of e.g.f. 1/(1 - sin(8*x))^(1/8).

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A230134 Expansion of e.g.f. 1/(1 - sin(5*x))^(1/5).

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A235128 Expansion of e.g.f. 1/(1 - sin(7*x))^(1/7).

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