A025266 -id:A025266 - cp's OEIS frontend

A214198 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index >= 3.

0, 0, 0, 2, 4, 12, 36, 116, 384, 1304, 4504, 15772, 55832, 199432, 717816, 2600680, 9476800, 34710000, 127712560, 471851180, 1749864920, 6511643720, 24307501720, 91000873560, 341594374400, 1285436348112, 4848292800336, 18325541062936, 69405260675824, 263353613108944, 1001028051476656, 3811242180811728, 14533071892504448

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.

Cf. A025266, A000108, A025271, A214198-A214205.

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

(1/2)*(Sqrt[1-4*x+8*x^3] - Sqrt[1-4*x]) + O[x]^33 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)

A214199 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 3.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 12, 36, 120, 392, 1288, 4284, 14304, 48024, 162024, 548872, 1866416, 6368464, 21797776, 74822636, 257513344, 888439192, 3072153864, 10645835384, 36964041872, 128584760560, 448087042160, 1564065659608, 5467992829120, 19144550862960, 67123334707984, 235658063191312, 828405764175712, 2915610778184352, 10273466501139232, 36239527330228044

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.

Cf. A025266, A000108, A025271, A214198-A214205.

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

(1/2)*(Sqrt[1-4*x+8*x^3]-Sqrt[1-4*x+16*x^4])+O[x]^36 // CoefficientList[#,x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)

A214200 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index <= 4.

Original entry on oeis.org

0, 1, 1, 2, 5, 6, 26, 84, 269, 870, 2910, 9788, 33250, 114012, 394260, 1372776, 4809917, 16947462, 60012470, 213462380, 762355286, 2732658484, 9827926060, 35453715480, 128255260690, 465163021788, 1691086242796, 6161413737176, 22494722099492, 82282062468600, 301507924857768, 1106652847697872, 4068159345287325, 14976738917364166

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.

Cf. A025266, A000108, A025271, A214198-A214205.

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

(1/2)*(1 - Sqrt[1 - 4*x + 32*x^5]) + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)

A214201 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index >= 4.

Original entry on oeis.org

0, 0, 0, 0, 4, 8, 24, 80, 264, 912, 3216, 11488, 41528, 151408, 555792, 2051808, 7610384, 28341536, 105914784, 397028544, 1492351576, 5623204528, 21235347856, 80355038176, 304630332528, 1156851587552, 4400205758176, 16761475403328, 63937267846704, 244209062245984, 933904716768672, 3575584117620416, 13704666128328736, 52582688861676096

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.

Cf. A025266, A000108, A025271, A214198-A214205.

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

(1/2)*(Sqrt[1-4*x+16*x^4]-Sqrt[1-4*x]) + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)

A214202 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 4.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 8, 32, 104, 352, 1264, 4480, 15992, 57408, 207152, 750144, 2725456, 9931328, 36282464, 132852224, 487443672, 1791742592, 6597006896, 24326190016, 89825979568, 332110462016, 1229345599520, 4555536068352, 16898439030192, 62743172964224, 233170424975072, 867250463225984, 3228189434389152, 12025362901992064, 44827564359795392

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.

Cf. A025266, A000108, A025271, A214198-A214205.

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

(1/2)*(Sqrt[1 - 4*x + 16*x^4] - Sqrt[1 - 4*x + 32*x^5]) + O[x]^35 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)

A214203 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index <= 5.

Original entry on oeis.org

0, 1, 1, 2, 5, 14, 26, 100, 333, 1110, 3742, 12764, 44258, 154636, 544660, 1932360, 6900029, 24780390, 89445174, 324326060, 1180834390, 4315287140, 15823305516, 58200045432, 214672363410, 793883691004, 2942917457772, 10933569255832, 40704185771812, 151826357818840, 567322837830824, 2123429246035600, 7960199797453213, 29884582184913542

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.
Filippo Disanto, Unbalanced subtrees in binary rooted ordered and un-ordered trees, Séminaire Lotharingien de Combinatoire, 68 (2013), Article B68b.

Cf. A025266, A000108, A025271, A214198-A214205.

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

(1/2)*(1 - Sqrt[1 - 4*x + 64*x^6]) + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)

A214204 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index >= 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 16, 48, 160, 560, 1952, 7008, 25536, 94000, 348640, 1301664, 4884928, 18410208, 69632320, 264176320, 1004907904, 3831461936, 14638340960, 56028848160, 214804352960, 824741125536, 3170860158656, 12205939334976, 47038828816512, 181465889281760, 700734291793600, 2708333654394432, 10476476693939584, 40557325959684032

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.
Filippo Disanto, Unbalanced subtrees in binary rooted ordered and un-ordered trees, Séminaire Lotharingien de Combinatoire, 68 (2013), Article B68b.

Cf. A025266, A000108, A025271, A214198-A214205.

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

(1/2)*(Sqrt[1-4*x+32*x^5] - Sqrt[1-4*x]) + O[x]^34 //CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)

A214205 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 0, 16, 64, 240, 832, 2976, 11008, 40624, 150400, 559584, 2090112, 7832928, 29432704, 110863680, 418479104, 1582628656, 5995379456, 22746329952, 86417102720, 328720669216, 1251831214976, 4772155518656, 18209463672320, 69544295350240, 265814912973056, 1016776398337728, 3892040452165888, 14907843267549376

Offset: 0

N. J. A. Sloane, Jul 07 2012

nonn

Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.
Filippo Disanto, Unbalanced subtrees in binary rooted ordered and un-ordered trees, Séminaire Lotharingien de Combinatoire, 68 (2013), Article B68b.

Cf. A025266, A000108, A025271, A214198-A214205.

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

(1/2)*(Sqrt[1 - 4*x + 32*x^5] - Sqrt[1 - 4*x + 64*x^6]) + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)

A257290 Number of 3-Motzkin paths of length n with no level steps at even level.

Original entry on oeis.org

1, 0, 1, 3, 11, 39, 140, 504, 1823, 6621, 24144, 88380, 324699, 1197045, 4427565, 16427385, 61129025, 228103185, 853399640, 3200710680, 12032399045, 45332769075, 171148151095, 647412581643, 2453529142471, 9314461044639, 35419207688050, 134894888442714, 514506926871927

Offset: 0

José Luis Ramírez Ramírez, Apr 20 2015

nonn

			For n=3 we have 3 paths: UH1D, UH2D, UH3D.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A090345, A025266.

CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

x='x+O('x^50); Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Feb 14 2017

a(n) = Sum_{i=0..floor(n/2)} 3^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the n-th Catalan number A000108.
G.f.: (1 - 3*z - sqrt((1-3*z)*(1-3*z-4*z^2)))/(2*z^2).
a(n) ~ sqrt(5) * 4^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
Conjecture: (n+2)*a(n) +3*(-2*n-1)*a(n-1) +5*(n-1)*a(n-2) +6*(2*n-5)*a(n-3)=0. - R. J. Mathar, Sep 24 2016

A025264 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)*a(1) for n >= 4, starting 2,1,1.

Original entry on oeis.org

2, 1, 1, 5, 22, 99, 450, 2067, 9586, 44852, 211570, 1005427, 4810460, 23157904, 112110906, 545524287, 2666864340, 13092764136, 64527778938, 319157531592, 1583724160896, 7882364163954, 39339994155288, 196843821874407, 987272738842392

Offset: 1

Clark Kimberling

nonn

Cf. A025266.

A025264 := proc(n)
    option remember ;
    if n < 4 then
        op(n,[2,1,1]) ;
    else
        add( procname(i)*procname(n-i),i=1..n-1) ;
    end if;
end proc:
seq(A025264(n),n=1..20) ; # R. J. Mathar, Jan 13 2025

nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 2; aa[[2]] = 1; aa[[3]] = 1; Do[aa[[n]] = Sum[aa[[k]] * aa[[n-k]],{k,1,n-1}],{n,4,nmax}]; aa (* Vaclav Kotesovec, Jan 25 2015 *)

a(n)=polcoeff((1-sqrt(1-8*x+12*x^2+12*x^3+x*O(x^n)))/2,n)

G.f.: (1-sqrt(1-8*x+12*x^2+12*x^3))/2. - Michael Somos, Jun 08 2000

Recurrence: n*a(n) = 4*(2*n-3)*a(n-1) - 12*(n-3)*a(n-2) - 6*(2*n-9)*a(n-3). - Vaclav Kotesovec, Jan 25 2015

cp's OEIS Frontend

A214198 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index >= 3.

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Maple

Mathematica

A214199 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 3.

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Maple

Mathematica

A214200 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index <= 4.

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Programs

Maple

Mathematica

A214201 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index >= 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A214202 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A214203 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index <= 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A214204 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index >= 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A214205 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A257290 Number of 3-Motzkin paths of length n with no level steps at even level.

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Author

Keywords

Examples

Links

Crossrefs

A025264 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)*a(1) for n >= 4, starting 2,1,1.