A006677 -id:A006677 - cp's OEIS frontend

A332776 a(n) = 1 + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

1, 1, 2, 5, 18, 83, 464, 3041, 22810, 192595, 1807328, 18658097, 210138882, 2563990859, 33691089824, 474327797585, 7123141539610, 113656386574099, 1920170741071280, 34242622099969217, 642792206343361602, 12669617513914228907, 261613287097165614224, 5647565141926833774977

Offset: 0

Ilya Gutkovskiy, Jun 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..454

Cf. A000110, A006677, A007317, A080635, A105633, A186021.

a[n_] := a[n] = 1 + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
terms = 23; A[] = 0; Do[A[x] = Normal[Integrate[Exp[x] + A[x] (A[x] - 1), x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

E.g.f. A(x) satisfies: d/dx A(x) = exp(x) + A(x) * (A(x) - 1).
From Vaclav Kotesovec, Jun 09 2020: (Start)
E.g.f.: exp(x/2) * (BesselJ(2, 2*exp(x/2)) * BesselY(0,2) - BesselJ(0,2) * BesselY(2, 2*exp(x/2))) / (BesselJ(1, 2*exp(x/2)) * BesselY(0,2) - BesselJ(0,2) * BesselY(1, 2*exp(x/2))).
a(n) ~ n! / r^(n+1), where r = 1.0654335847261788612657252860730850911833168584... is the smallest real root of the equation BesselJ(1, 2*exp(r/2)) * BesselY(0,2) = BesselJ(0,2) * BesselY(1, 2*exp(r/2)). (End)

A345102 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 4, 37, 589, 13276, 386059, 13741057, 578451514, 28109736811, 1548565036789, 95365652263102, 6492034471389889, 484086370908869821, 39238367740327468444, 3435176518078688461297, 323029539924876486293089, 32472511993953383052630556, 3475005417300807667690138399

Offset: 0

Ilya Gutkovskiy, Jun 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A006677, A011781, A052886, A201354, A345103.

a[n_] := a[n] = 1 + 3 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[7 - 6 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 3^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]

N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(7-6*exp(x)))) \\ Seiichi Manyama, Oct 20 2021

E.g.f.: exp(x) / sqrt(7 - 6 * exp(x)).

A346119 Expansion of the e.g.f. sqrt(2xexp(x) - 2*exp(x) + 3).

Original entry on oeis.org

1, 0, 1, 2, 0, -16, -35, 342, 2779, -6424, -239382, -822460, 22393657, 278844084, -1553468891, -68399947042, -275025888900, 15302175612416, 243541868882077, -2463105309082902, -121649966081262521, -473088821582805820, 50905612811064360006, 945133249101683013812, -15321255878414345388335

Offset: 0

Mélika Tebni, Jul 05 2021

sign

			sqrt(2*x*exp(x)-2*exp(x)+3) = 1 + x^2/2! + 2*x^3/3! - 16*x^5/5! - 35*x^6/6! + 342*x^7/7! + 2779*x^8/8! - 6424*x^9/9! + ...
a(11) = Sum_{k=1..5} (-1)^(k-1)*A006677(k)*A008306(11,k) = -822460.
For k=1, (-1)^(1-1)*A006677(1)*A008306(11,1) == -1 (mod 11), because A006677(1) = 1 and A008306(11,1) = (11-1)!
For k>=2, (-1)^(k-1)*A006677(k)*A008306(11,k) == 0 (mod 11), because A008306(11,k) == 0 (mod 11), result a(11) == -1 (mod 11).
a(8) = Sum_{k=1..4} (-1)^(k-1)*A006677(k)*A008306(8,k) = 2779.
a(8) == 0 (mod (8-1)), because for k >= 1, A008306(8,k) == 0 (mod 7).

Cf. A000166, A006677, A008306, A327006, A345652, A345697, A345969.

stirtr:= proc(p) proc(n) add(p(k)*Stirling2(n, k), k=0..n) end end: f:= n-> `if`(n=0, 1, (2*n-2)!/ (n-1)!/ 2^(n-1)): A006677:= stirtr(f): # Alois P. Heinz, 2008.
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a:= n-> add(((-1)^(k-1)*A006677(k)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(sqrt(2*x*exp(x)-2*exp(x)+3), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);

CoefficientList[Series[Sqrt(2*x*E^x-2*E^x+3), {x, 0, 24}], x] * Range[0, 24]!

my(x='x+O('x^30)); Vec(serlaplace(sqrt(2*x*exp(x) - 2*exp(x) + 3))) \\ Michel Marcus, Jul 05 2021

E.g.f. y(x) satisfies y*y' = x*exp(x).
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1)*A006677(k)*A008306(n,k) for n > 0.
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
Conjecture: a(n) = 0 for only n = 1 and n = 4.

A335441 a(n) = 1 + Sum_{k=1..n-1} binomial(n-2,k-1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 2, 4, 11, 40, 176, 907, 5360, 35668, 263789, 2146390, 19054040, 183248581, 1897952690, 21061861828, 249309196559, 3135518918800, 41754612283244, 586922460056851, 8684272948653068, 134919751191875572, 2195942678525060093, 37365571515146318650

Offset: 0

Ilya Gutkovskiy, Jun 10 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..470

Cf. A000111, A006677, A052886, A054687, A080635, A332776.

a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k - 1] a[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
terms = 23; A[] = 0; Do[A[x] = Normal[Integrate[Integrate[Exp[x] + A[x] D[A[x], x], x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

E.g.f. A(x) satisfies: A''(x) = exp(x) + A(x) * A'(x).
From Vaclav Kotesovec, Jun 11 2020: (Start)
E.g.f.: (BesselY(0, sqrt(2))*(BesselJ(1, sqrt(2)*exp(x/2)) - sqrt(2)*exp(x/2)*BesselJ(0, sqrt(2)*exp(x/2))) + BesselJ(0, sqrt(2))*(sqrt(2)*exp(x/2)*BesselY(0, sqrt(2)*exp(x/2)) - BesselY(1, sqrt(2)*exp(x/2)))) / (BesselJ(1, sqrt(2)*exp(x/2))*BesselY(0, sqrt(2)) - BesselJ(0, sqrt(2))*BesselY(1, sqrt(2)*exp(x/2))).
a(n) ~ 2 * n! / r^(n+1), where r = 1.35169030867903432729790416904526340210784862703704233748118252928787... is the smallest real root of the equation BesselY(0, sqrt(2))*BesselJ(1, sqrt(2)*exp(r/2)) = BesselJ(0, sqrt(2))*BesselY(1, sqrt(2)*exp(r/2)). (End)

A345103 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 5, 61, 1277, 37741, 1437725, 67013101, 3693540317, 234974905261, 16945434018845, 1366008048556141, 121721015465713757, 11880107754103150381, 1260413749895624939165, 144427420001275864755181, 17776090894283922227621597, 2338833689096321086977341101, 327585830473259220341296486685

Offset: 0

Ilya Gutkovskiy, Jun 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..331

Cf. A006677, A052886, A144828, A201365, A345102.

a[n_] := a[n] = 1 + 4 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[9 - 8 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 4^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]

N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(9-8*exp(x)))) \\ Seiichi Manyama, Oct 20 2021

E.g.f.: exp(x) / sqrt(9 - 8 * exp(x)).

A006678 Number of planted binary phylogenetic trees with n labels.

Original entry on oeis.org

0, 1, 1, 6, 39, 390, 4815, 73080, 1304415, 26847450, 625528575, 16279193700, 468022452975, 14731683916950, 503880434632575, 18609309606888000, 738057377647965375, 31286068076704691250, 1411602439003747365375, 67543694966494692445500, 3416288904242387117319375

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Foulds, L. R.; Robinson, R. W. Enumeration of binary phylogenetic trees. Combinatorial mathematics, VIII (Geelong, 1980), pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. Math. Rev. 83a:05071.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. R. Foulds and R. W. Robinson, Enumeration of binary phylogenetic trees, pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. (Annotated scanned copy)
Plouffe, Simon, Master's Thesis, Uqam, 1992.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

CoefficientList[Series[-(-1+(1-2*x-2*x^2)^(1/2))/(1+x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

E.g.f.: -(-1+(1-2*x-2*x^2)^(1/2))/(1+x), Simon Plouffe in his Master's Thesis, Uqam 1992.

a(n) ~ sqrt(6-2*sqrt(3)) * (1+sqrt(3))^(n-1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 01 2013

A006679 Number of planted binary phylogenetic trees with n labels.

Original entry on oeis.org

0, 1, 2, 10, 83, 946, 13772, 244315, 5113208, 123342166, 3369568817, 102831001120, 3467225430308, 128006254663561, 5135734326127862, 222498607683528550, 10352141336173618883, 514815201125079097006, 27251316075035460318032, 1529842445683171428183355

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 0..200
L. R. Foulds, R. W. Robinson, Enumeration of binary phylogenetic trees Combinatorial mathematics, VIII (Geelong, 1980), pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. Math. Rev. 83a:05071.
L. R. Foulds and R. W. Robinson, Enumeration of binary phylogenetic trees, pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. (Annotated scanned copy)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

CoefficientList[Series[(1 - Sqrt[1 + 2*E^x - 2*E^(2*x)])/E^x, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Nov 29 2015 *)

a(n):=sum(((sum((2^(1-k)*(2*k-2)!/(k-1)!*sum((stirling2(i,k)*k^(j-i))*binomial(j,i),i,0,j)),k,1,j))*(-1)^(n-j))*binomial(n,j),j,0,n); /* Vladimir Kruchinin, Nov 17 2015 */

x='x+O('x^33); Vec(serlaplace((1 - (1 + 2*exp(x) - 2*exp(x)^2)^(1/2))/exp(x))) \\ Gheorghe Coserea, Aug 04 2015

E.g.f: (1 - (1 + 2*exp(x) - 2*exp(x)^2)^(1/2))/exp(x). - Gheorghe Coserea, Aug 04 2015
a(n) = Sum_{j=0..n} C(n,j)*(-1)^(n-j)* Sum_{k=1..j} 2^(1-k)*(2*k-2)!/ (k-1)!* Sum_{i=0..j} stirling2(i,k)*k^(j-i)*C(j,i). - Vladimir Kruchinin, Nov 17 2015
a(n) ~ sqrt(3-sqrt(3)) * n^(n-1) / (log((1+sqrt(3))/2)^(n-1/2) * exp(n)). - Vaclav Kotesovec, Nov 29 2015

More terms from Gheorghe Coserea, Aug 04 2015

A006680 Number of binary phylogenetic trees with n labels.

Original entry on oeis.org

1, 1, 7, 45, 465, 5775, 88515, 1588545, 32852925, 768242475, 20053670175, 577971719325, 18230691904425, 624676513836375, 23106221910646875, 917643290676737625, 38944796393877499125, 1759004961887566267875

Offset: 2

N. J. A. Sloane

nonn

Foulds, L. R.; Robinson, R. W. Enumeration of binary phylogenetic trees. Combinatorial mathematics, VIII (Geelong, 1980), pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. Math. Rev. 83a:05071.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. R. Foulds and R. W. Robinson, Enumeration of binary phylogenetic trees, pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. (Annotated scanned copy)
Index entries for sequences related to trees

a(2) = 1, a(n) = A006678(n-1) + 2*(n-1)*A006678(n-2) - (n+1)*a(n-1) for n >= 3. - Sean A. Irvine, Jun 16 2017

More terms and offset corrected by Sean A. Irvine, Jun 16 2017

A006681 Number of binary phylogenetic trees with n labels.

Original entry on oeis.org

1, 4, 16, 85, 646, 6664, 86731, 1354630, 24607816, 509000899, 11802833296, 303129872500, 8539567141561, 261785254856584, 8674697402099836, 308953306192527445, 11768768651529113506, 477433786852228938844

Offset: 2

N. J. A. Sloane

nonn

Foulds, L. R.; Robinson, R. W. Enumeration of binary phylogenetic trees. Combinatorial mathematics, VIII (Geelong, 1980), pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. Math. Rev. 83a:05071.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. R. Foulds and R. W. Robinson, Enumeration of binary phylogenetic trees, pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. (Annotated scanned copy)
Index entries for sequences related to trees

a(n) = Sum_{k=2..n} S(n, k) * A001147(k - 2) where S(n, m) are the Stirling numbers of the second kind. - Sean A. Irvine, Jun 16 2017

E.g.f.: ( -4 + 3*exp(x) + (3 - 2*exp(x))^(3/2) )/3. (see (3.1) in the L. R. Foulds and R. W. Robinson reference). - Seiichi Manyama, Sep 05 2024

More terms and offset corrected by Sean A. Irvine, Jun 16 2017

A309473 a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 0, 1, 1, 3, 11, 43, 195, 1063, 6395, 42371, 311883, 2501159, 21672355, 202544323, 2028522067, 21658255431, 245738583307, 2952508103651, 37440976938875, 499785548010759, 7005210659040979, 102862231664567651, 1579045889274408259, 25294106622048460903

Offset: 0

Ilya Gutkovskiy, Jun 11 2020

nonn

Cf. A006677, A014304, A054687, A332776, A335441.

a[n_] := a[n] = (-1)^n + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
terms = 24; A[] = 1; Do[A[x] = Normal[Integrate[A[x]^2 - Exp[-x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

E.g.f. A(x) satisfies: A'(x) = A(x)^2 - exp(-x).
From Vaclav Kotesovec, Jun 11 2020: (Start)
E.g.f.: exp(-x/2) * (BesselI(1, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2)) + (BesselI(0, 2) - BesselI(1, 2)) * BesselK(1, 2*exp(-x/2))) / ((BesselI(1, 2) - BesselI(0, 2)) * BesselK(0, 2*exp(-x/2)) + BesselI(0, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2))).
a(n) ~ n! / r^(n+1), where r = 1.4982609322383959128764444062824740935658895762... is the real root of the equation (BesselI(0, 2) - BesselI(1, 2)) * BesselK(0, 2*exp(-r/2)) = (BesselK(0, 2) + BesselK(1, 2)) * BesselI(0, 2*exp(-r/2)). (End)

cp's OEIS Frontend

A332776 a(n) = 1 + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

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A345102 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

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A346119 Expansion of the e.g.f. sqrt(2*x*exp(x) - 2*exp(x) + 3).

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A335441 a(n) = 1 + Sum_{k=1..n-1} binomial(n-2,k-1) * a(k) * a(n-k-1).

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A345103 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

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A006678 Number of planted binary phylogenetic trees with n labels.

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A006679 Number of planted binary phylogenetic trees with n labels.

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Maxima

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A006680 Number of binary phylogenetic trees with n labels.

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A006681 Number of binary phylogenetic trees with n labels.

A346119 Expansion of the e.g.f. sqrt(2xexp(x) - 2*exp(x) + 3).