A239108 -id:A239108 - cp's OEIS frontend

A245049 Number A(n,k) of hybrid k-ary trees with n internal nodes; square array A(n,k), n>=0, k>=1, read by antidiagonals.

1, 1, 2, 1, 2, 3, 1, 2, 7, 5, 1, 2, 11, 31, 8, 1, 2, 15, 81, 154, 13, 1, 2, 19, 155, 684, 820, 21, 1, 2, 23, 253, 1854, 6257, 4575, 34, 1, 2, 27, 375, 3920, 24124, 60325, 26398, 55, 1, 2, 31, 521, 7138, 66221, 331575, 603641, 156233, 89, 1, 2, 35, 691, 11764, 148348, 1183077, 4736345, 6210059, 943174, 144

Offset: 0

Alois P. Heinz, Jul 10 2014

			Square array A(n,k) begins:
   1,    1,     1,      1,       1,       1,       1, ...
   2,    2,     2,      2,       2,       2,       2, ...
   3,    7,    11,     15,      19,      23,      27, ...
   5,   31,    81,    155,     253,     375,     521, ...
   8,  154,   684,   1854,    3920,    7138,   11764, ...
  13,  820,  6257,  24124,   66221,  148348,  290305, ...
  21, 4575, 60325, 331575, 1183077, 3262975, 7585749, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235

Columns k=1-10 give: A000045(n+2), A007863, A215654, A239107, A239108, A239109, A245050, A245051, A245052, A245053.
Rows n=0-2 give: A000012, A007395, A004767(k-1).
Main diagonal gives A245054.

A:= (n, k)-> add(binomial((k-1)*n+i, i)*
    binomial((k-1)*n+i+1, n-i), i=0..n)/((k-1)*n+1):
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);

A[n_, k_] := Sum[Binomial[(k-1)*n+i, i]*Binomial[(k-1)*n+i+1, n-i], {i, 0, n}]/((k-1)*n+1); Table[A[n, 1+d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)

A(n,k) = 1/((k-1)*n+1) * Sum_{i=0..n} C((k-1)*n+i,i)*C((k-1)*n+i+1,n-i).
A(n,k) = [x^n] ((1+x)/(1-x-x^2))^((k-1)*n+1) / ((k-1)*n+1).
G.f. for column k satisfies: A_k(x) = (1+x*A_k(x)^(k-1)) * (1+x*A_k(x)^k).

A364338 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 11, 105, 1140, 13555, 170637, 2235472, 30161255, 416248640, 5848462880, 83378361111, 1203100853951, 17537182300140, 257858115407535, 3819894878557990, 56958234329850060, 854192593184162160, 12875579347191388830, 194963091634569681550, 2964229359714424159370, 45234864131654311730160

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..827

Cf. A073157, A364336, A364337, A364339.

Cf. A215624, A234525, A239108, A364331, A364335.

terms = 22; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^5) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)

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

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

A239109 Number of hybrid 6-ary trees with n internal nodes.

Original entry on oeis.org

1, 2, 23, 375, 7138, 148348, 3262975, 74673216, 1759690865, 42412172598, 1040644972314, 25907046248766, 652763779424538, 16614703783094140, 426563932954831827, 11033640140115676862, 287265076610919864178, 7522060666571155198520, 197969862318742854908470

Offset: 0

N. J. A. Sloane, Mar 26 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300
SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235. See p. 233.

Cf. A000045, A007863, A215654, A239107, A239108, A239109.

Column k=6 of A245049.

(1/x InverseSeries[x*(1 - x - x^2)^5/(1 + x)^5 + O[x]^20])^(1/5) // CoefficientList[#, x]& (* Jean-François Alcover, Oct 02 2019 *)

a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1 + x*A^5)*(1 + x*A^6)); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

a(n)=polcoeff( ((1/x)*serreverse( x*(1-x-x^2)^5/(1+x +x*O(x^n))^5))^(1/5), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

a(n)=local(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*A^j)*x^m*A^(4*m)/m))); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

a(n)=local(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/A^j)*x^m*A^(5*m)/m))); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

a(n)=polcoeff(((1+x)/(1-x-x^2 +x*O(x^n)))^(5*n+1)/(5*n+1), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

From Paul D. Hanna, Mar 30 2014: (Start)
G.f. A(x) satisfies:
(1) A(x) = (1 + x*A(x)^5) * (1 + x*A(x)^6).
(2) A(x) = ( (1/x)*Series_Reversion( x*(1-x-x^2)^5/(1+x)^5 ) )^(1/5).
(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(4*n)/n * Sum_{k=0..n} C(n,k)^2 * A(x)^k ).
(4) A(x) = exp( Sum_{n>=1} x^n*A(x)^(5*n)/n * Sum_{k=0..n} C(n,k)^2 / A(x)^k ).
(5) A(x) = Sum_{n>=0} Fibonacci(n+2) * x^n * A(x)^(5*n).
(6) A(x) = G(x*A(x)^4) where G(x) = A(x/G(x)^4) is the g.f. of A007863 (number of hybrid binary trees with n internal nodes).
The formal inverse of g.f. A(x) is (sqrt(1-2*x+5*x^2) - (1+x))/(2*x^6).
a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(5*n+1) / (5*n+1).
(End)

A239107 Number of hybrid 4-ary trees with n internal nodes.

Original entry on oeis.org

1, 2, 15, 155, 1854, 24124, 331575, 4736345, 69616637, 1046054129, 15995716263, 248111418112, 3894303176880, 61737213540306, 987116931080661, 15899835212249761, 257758369219909534, 4202381519278498915, 68859442092723799788, 1133401910867109123200

Offset: 0

N. J. A. Sloane, Mar 26 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..800
SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235. See p. 233.
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)

Cf. A000045, A007863, A215654, A239107, A239108, A239109.

Column k=4 of A245049.

(1/x InverseSeries[x(1 - x - x^2)^3/(1 + x)^3 + O[x]^21])^(1/3) // CoefficientList[#, x]& (* Jean-François Alcover, Oct 02 2019 *)

a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1 + x*A^3)*(1 + x*A^4)); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

a(n)=polcoeff( ((1/x)*serreverse( x*(1-x-x^2)^3/(1+x +x*O(x^n))^3))^(1/3), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

a(n)=local(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*A^j)*x^m*A^(2*m)/m))); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

a(n)=local(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/A^j)*x^m*A^(3*m)/m))); polcoeff(A, n) \\ Paul D. Hanna, Mar 30 2014
for(n=0, 20, print1(a(n), ", "))

a(n)=polcoeff(((1+x)/(1-x-x^2 +x*O(x^n)))^(3*n+1)/(3*n+1), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

From Paul D. Hanna, Mar 30 2014: (Start)
G.f. A(x) satisfies:
(1) A(x) = (1 + x*A(x)^3) * (1 + x*A(x)^4).
(2) A(x) = ( (1/x)*Series_Reversion( x*(1-x-x^2)^3/(1+x)^3 ) )^(1/3).
(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(2*n)/n * Sum_{k=0..n} C(n,k)^2 * A(x)^k ).
(4) A(x) = exp( Sum_{n>=1} x^n*A(x)^(3*n)/n * Sum_{k=0..n} C(n,k)^2 / A(x)^k ).
(5) A(x) = Sum_{n>=0} Fibonacci(n+2) * x^n * A(x)^(3*n).
(6) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A007863 (number of hybrid binary trees with n internal nodes).
The formal inverse of g.f. A(x) is (sqrt(1-2*x+5*x^2) - (1+x))/(2*x^4).
a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(3*n+1) / (3*n+1).
(End)
a(n) = 1/(3*n+1) * Sum_{i=0..n} C(3*n+i,i)*C(3*n+i+1,n-i). - Alois P. Heinz, Jul 10 2014

A364331 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 15, 163, 2070, 28698, 421015, 6425644, 100977137, 1622885389, 26551709946, 440744175801, 7404449354076, 125657625548824, 2150963575012295, 37094953102567208, 643904274979347286, 11241232087809137759, 197247501440314516840, 3476787208220672891388, 61533794803235280779261

Offset: 0

Seiichi Manyama, Jul 18 2023

Cf. A007863, A069271, A073157, A215654, A215715, A364333.
Cf. A215623, A215624, A239108, A364335, A364338.
Cf. A200719.

A364331 := proc(n)
    add( binomial(2*n+3*k+1,k) * binomial(2*n+3*k+1,n-k)/(2*n+3*k+1),k=0..n) ;
end proc:
seq(A364331(n),n=0..70); # R. J. Mathar, Jul 25 2023

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

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

x/series_reversion(x*A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + ..., the g.f. of A215623. - Peter Bala, Sep 08 2024

A364335 G.f. satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 17, 204, 2852, 43489, 701438, 11767095, 203223146, 3589167533, 64524575635, 1176860764416, 21723084076739, 405038036077647, 7617437252889030, 144328483391622298, 2752414654270742784, 52790626691557217602, 1017655117382823639414, 19706520281177438174530

Offset: 0

Seiichi Manyama, Jul 18 2023

nonn

Cf. A215624, A234525, A239108, A364331, A364338.

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

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

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).

cp's OEIS Frontend

A245049 Number A(n,k) of hybrid k-ary trees with n internal nodes; square array A(n,k), n>=0, k>=1, read by antidiagonals.

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A364338 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^5).

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A239109 Number of hybrid 6-ary trees with n internal nodes.

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A239107 Number of hybrid 4-ary trees with n internal nodes.

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A364331 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^5).

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A364335 G.f. satisfies A(x) = (1 + x*A(x)^3) * (1 + x*A(x)^5).

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

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A364331 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^5).

A364335 G.f. satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x)^5).