A200753 -id:A200753 - cp's OEIS frontend

A346627 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * A(x)^3.

1, 0, 1, 2, 7, 23, 82, 300, 1129, 4334, 16914, 66899, 267586, 1080516, 4398850, 18035084, 74402361, 308624282, 1286428765, 5385578256, 22635057148, 95471113565, 403983783772, 1714494024947, 7295949019114, 31124885587680, 133085594104222, 570266646942488

Offset: 0

Ilya Gutkovskiy, Jul 25 2021

nonn

Inverse binomial transform of A200753.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001764, A005043, A199475, A200753, A346628, A349299, A349300, A349301, A349302, A349303.

nmax = 27; A[] = 0; Do[A[x] = 1/(1 + x) + x A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 27}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 27}]

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 + x)^(2*k+1).
a(n) = (-1)^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * binomial(3*k,k) / (2*k + 1).
a(n) ~ sqrt(198 + 38*sqrt(33)) * (19 + 3*sqrt(33))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 3)). - Vaclav Kotesovec, Jul 30 2021

A200754 G.f. satisfies A(x) = 1 + xA(x)^4 - x^2A(x)^5.

Original entry on oeis.org

1, 1, 3, 13, 67, 380, 2288, 14351, 92737, 613063, 4126289, 28179766, 194780822, 1360053081, 9578997279, 67971291791, 485464864401, 3487203531460, 25176899072984, 182598098616625, 1329716528758651, 9718954060263384, 71273846758123552, 524279847227139350

Offset: 0

Paul D. Hanna, Nov 21 2011

nonn

Compare to the g.f. G(x) for the ternary tree numbers (A001764): G(x) = 1 + x*G(x)^4 - x^2*G(x)^6 = 1 + x*G(x)^3.

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 67*x^4 + 380*x^5 + 2288*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 18*x^2 + 92*x^3 + 515*x^4 + 3068*x^5 + 19092*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 780*x^4 + 4741*x^5 + 29915*x^6 +...
where a(2) = 4 - 1; a(3) = 18 - 5; a(4) = 92 - 25; a(5) = 515 - 135; ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A200753, A200755, A365180.

{a(n)=local(A=1+x);for(i=1,n,A=1+x*A^4-x^2*A^5+x*O(x^n));polcoeff(A,n)}

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n+3*k+1, k)*binomial(k, n-k)/(n+3*k+1)); \\ Seiichi Manyama, Nov 01 2023

Recurrence: 3*n*(n+1)*(3*n - 1)*(3*n + 1)*(1280*n^7 - 17280*n^6 + 94466*n^5 - 266799*n^4 + 407516*n^3 - 311946*n^2 + 81648*n + 12420)*a(n) = n*(281600*n^10 - 3942400*n^9 + 22465080*n^8 - 65757900*n^7 + 98435187*n^6 - 46293273*n^5 - 73831865*n^4 + 131625925*n^3 - 85490202*n^2 + 25470288*n - 2915460)*a(n-1) + (263680*n^11 - 4087040*n^10 + 26682396*n^9 - 96110406*n^8 + 210443037*n^7 - 293037231*n^6 + 270493529*n^5 - 181862299*n^4 + 103208358*n^3 - 47593224*n^2 + 12881700*n - 1360800)*a(n-2) - 5*(640000*n^11 - 10880000*n^10 + 78609000*n^9 - 311686500*n^8 + 721945299*n^7 - 918913929*n^6 + 367548335*n^5 + 642045653*n^4 - 1081692258*n^3 + 697174344*n^2 - 208955268*n + 24222240)*a(n-3) + 5*(5*n - 18)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(1280*n^7 - 8320*n^6 + 17666*n^5 - 8869*n^4 - 15820*n^3 + 22148*n^2 - 9282*n + 1305)*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ s*sqrt((2*r*s-1) / (2*Pi*(5*r*s-3))) / (2*n^(3/2)*r^n), where r = 0.1272568969777848138753091632571986265610307654216... and s = 1.358291097397172238669759690645074441686961930838... are roots of the system of equations s + r^2*s^5 = 1 + r*s^4, 1 + 5*r^2*s^4 = 4*r*s^3. - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+3*k+1,k) * binomial(k,n-k)/(n+3*k+1). - Seiichi Manyama, Nov 01 2023

A137265 G.f. y(x) is solution of x y^3 - (1 + x^2) y + 1 = 0 with y(0) = 1.

Original entry on oeis.org

1, 1, 2, 8, 35, 163, 796, 4024, 20885, 110654, 596064, 3254752, 17974893, 100227022, 563482140, 3190633232, 18179765509, 104158703503, 599698459613, 3467978715612, 20134256546896, 117313279477959, 685756774642494, 4020515276730588, 23636036336651811

Offset: 0

Robert Israel, Mar 12 2008

			a(3) = 8 because g(x) = 1 + x + 2 x^2 + 8 x^3 + O(x^4) satisfies x*g(x)^3 - (1 + x^2)*g(x) + 1 = O(x^4).

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000108, A200753, A200755.

Cf. A364474.

f:= (x,y) -> x*y^3 - (1 + x^2)*y + 1; N:= (y,n) -> convert(normal(taylor(y-f(x,y)/D[2](f)(x,y),x=0,n)),polynom); Y:= 1; for j from 1 to 6 do Y:= N(Y,2^j) end do; seq(coeftayl(Y,x=0,j),j=0..2^6-1);

max = 22; g[x_] := Sum[a[k]*x^k, {k, 0, max}]; coes = CoefficientList[ Series[ x*g[x]^3 - (1+x^2)*g[x] + 1, {x, 0, max}], x]; sol = First[ Solve[ Thread[ coes == 0 ] ] ]; Table[a[n] /. sol, {n, 0, max}](* Jean-François Alcover, Nov 28 2011 *)
terms = 25; y[] = 1; Do[y[x] = (1 + x*y[x]^3)/(1 + x^2) + O[x]^terms, terms]; CoefficientList[y[x], x] (* Jean-François Alcover, Jan 11 2018 *)

a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1)); \\ Seiichi Manyama, Nov 02 2023

a(0) = 1, a(1) = 1, a(n) = -a(n-2) + sum_{i=0}^{n-1} sum_{j=0}^{n-1-i} a(i) a(j) a(n-1-i-j).
a(n) ~ sqrt(1 - (2*r)^(5/3)) / (2^(4/3) * sqrt(3*Pi) * n^(3/2) * r^(n + 1/3)), where r = 0.15978798947663136723274504893788499231133813071845... is the real root of the equation (1+r^2)^3 = 27*r/4. - Vaclav Kotesovec, May 03 2016
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k) / (2*n-4*k+1). - Seiichi Manyama, Nov 02 2023

A200755 G.f. satisfies A(x) = 1 + xA(x)^3 - x^2A(x)^2.

Original entry on oeis.org

1, 1, 2, 7, 29, 129, 602, 2910, 14447, 73234, 377487, 1972568, 10425930, 55640282, 299403552, 1622701202, 8850030065, 48534971244, 267486182192, 1480673755443, 8228819436898, 45895682480965, 256815165790211, 1441321638029496, 8111194646903282

Offset: 0

Paul D. Hanna, Nov 21 2011

nonn

Compare to the g.f. C(x) for the Catalan numbers (A000108): C(x) = 1 + x*C(x)^3 - x^2*C(x)^4 = 1 + x*C(x)^2.

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 129*x^5 + 602*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 344*x^5 + 1627*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 678*x^5 + 3254*x^6 +...
where a(2) = 3 - 1; a(3) = 9 - 2; a(4) = 34 - 5; a(5) = 147 - 18; ...

Cf. A104979, A200754.

Cf. A000108, A137265, A200753.

nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[1 + x*AGF^3 - x^2*AGF^2 - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *)

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

a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-4*k, k)*binomial(3*n-5*k, n-2*k)/(2*n-3*k+1)); \\ Seiichi Manyama, Nov 02 2023

Recurrence: 2*n*(2*n+1)*(244*n^3 - 1713*n^2 + 3767*n - 2550)*a(n) = 3*(2196*n^5 - 17613*n^4 + 49628*n^3 - 59841*n^2 + 30478*n - 5184)*a(n-1) - 18*(244*n^5 - 2323*n^4 + 8013*n^3 - 12252*n^2 + 7774*n - 1260)*a(n-2) - (n-4)*(244*n^4 - 1713*n^3 + 4172*n^2 - 3333*n - 378)*a(n-3) - 36*(n-5)*(n-3)*(5*n + 2)*a(n-4) - 4*(n-6)*(n-4)*(244*n^3 - 981*n^2 + 1073*n - 252)*a(n-5). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 5.991151107674316485... is the root of the equation -4 - 4*d - 5*d^2 - 23*d^3 + 4*d^4 = 0 and c = 0.214566307956522153666714736272121899143... - Vaclav Kotesovec, Sep 10 2013
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-4*k,k) * binomial(3*n-5*k,n-2*k) / (2*n-3*k+1). - Seiichi Manyama, Nov 02 2023

A367017 G.f. satisfies A(x) = 1 + xA(x)^5(1 - x).

Original entry on oeis.org

1, 1, 4, 25, 185, 1495, 12776, 113534, 1038535, 9713905, 92480570, 893215584, 8730601596, 86198356180, 858388634250, 8611765147660, 86958794304735, 883103159075400, 9013769253136005, 92419535419392485, 951446700812718515, 9831013564639954705

Offset: 0

Seiichi Manyama, Nov 01 2023

nonn

Cf. A200753, A367016.

Cf. A002294, A365184.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k, n-k)*binomial(5*k, k)/(4*k+1));

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k,n-k) * A002294(k).

A374553 Number of length n inversion sequences avoiding the patterns 010 and 102.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 186, 707, 2763, 11024, 44714, 183830, 764374, 3209031, 13584217, 57918257, 248502212, 1072159593, 4648747281, 20245772943, 88524364619, 388469248937, 1710304847176, 7552480937589, 33442335151831, 148456424569164, 660560252794208

Offset: 0

Benjamin Testart, Jul 17 2024

nonn

Benjamin Testart, Table of n, a(n) for n = 0..700
Jay Pantone, The enumeration of inversion sequences avoiding the patterns 201 and 210, arXiv:2310.19632 [math.CO], 2023.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.
Benjamin Testart, Generating trees growing on the left for pattern-avoiding inversion sequences, arXiv:2411.05726 [math.CO], 2024.

Cf. A200753, A263779, A279559, A279566, A374541, A374542.

G.f. F(x) is algebraic with minimal polynomial x * (x^2 - x + 1)*(x - 1)^2 * F(x)^3 + 2*x*(x - 1)*(2*x^2 - 2*x + 1)*F(x)^2 - (x^4 - 8*x^3 + 11*x^2 - 6*x + 1)*F(x) - (2*x - 1)*(x - 1)^2.

A367016 G.f. satisfies A(x) = 1 + xA(x)^4(1 - x).

Original entry on oeis.org

1, 1, 3, 14, 78, 475, 3057, 20446, 140702, 989789, 7085635, 51451482, 378049810, 2805616460, 20999408480, 158337719608, 1201585477436, 9170328295222, 70339328959266, 541953619822062, 4192560258116202, 32552250308843605, 253583917423039079

Offset: 0

Seiichi Manyama, Nov 01 2023

nonn

Cf. A200753, A367017.

Cf. A002293, A365178.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k, n-k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k,n-k) * A002293(k).

A367027 G.f. A(x) satisfies A(x) = 1 + xA(x)^3 - x^2A(x)^5.

Original entry on oeis.org

1, 1, 2, 4, 5, -13, -147, -816, -3534, -12650, -35420, -53040, 199056, 2391340, 14555740, 68264112, 261045693, 769660569, 1167906402, -5145668100, -61758940705, -385813067255, -1857144860445, -7266981925560, -21793022441775, -32643056947527, 161919845140752

Offset: 0

Seiichi Manyama, Nov 02 2023

sign

Cf. A000108, A137265, A200753, A200755.

Cf. A361245, A368969.

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

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

G.f.: ( (1/x) * Series_Reversion( x * (1-x+x^2)^2 ) )^(1/2). - Seiichi Manyama, Mar 08 2025

A374554 Number of length n inversion sequences avoiding the patterns 100 and 102.

Original entry on oeis.org

1, 1, 2, 6, 21, 80, 318, 1305, 5487, 23535, 102603, 453400, 2026408, 9144361, 41607161, 190675552, 879318056, 4077566276, 19001732690, 88940105945, 417948841012, 1971086634986, 9326180071850, 44258248464408, 210605264950063, 1004694354945863, 4804017049287049

Offset: 0

Benjamin Testart, Jul 17 2024

nonn

Benjamin Testart, Table of n, a(n) for n = 0..400
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.

Cf. A200753, A263780, A279564, A279566, A374541, A374542, A374544, A374545, A374553.

cp's OEIS Frontend

A346627 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * A(x)^3.

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A200754 G.f. satisfies A(x) = 1 + x*A(x)^4 - x^2*A(x)^5.

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A137265 G.f. y(x) is solution of x y^3 - (1 + x^2) y + 1 = 0 with y(0) = 1.

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A200755 G.f. satisfies A(x) = 1 + x*A(x)^3 - x^2*A(x)^2.

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A367017 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 - x).

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A374553 Number of length n inversion sequences avoiding the patterns 010 and 102.

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A367016 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 - x).

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A367027 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 - x^2*A(x)^5.

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A374554 Number of length n inversion sequences avoiding the patterns 100 and 102.

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A200754 G.f. satisfies A(x) = 1 + xA(x)^4 - x^2A(x)^5.

A200755 G.f. satisfies A(x) = 1 + xA(x)^3 - x^2A(x)^2.

A367017 G.f. satisfies A(x) = 1 + xA(x)^5(1 - x).

A367016 G.f. satisfies A(x) = 1 + xA(x)^4(1 - x).

A367027 G.f. A(x) satisfies A(x) = 1 + xA(x)^3 - x^2A(x)^5.