cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A219537 G.f. satisfies A(x) = 1 + x*(A(x)^2 - A(x)^3 + A(x)^4).

Original entry on oeis.org

1, 1, 3, 13, 66, 366, 2148, 13115, 82449, 530095, 3469401, 23037642, 154820262, 1050999343, 7196493255, 49644745965, 344704716018, 2407157839593, 16895247295947, 119121868831235, 843306880720218, 5992060655349521, 42718501097385207, 305476181765843358
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2012

Keywords

Comments

a(n) is the number of noncrossing partial matchings on points 1, 2, ... , 3*n where point 1 is unmatched if n>0 and only points congruent modulo 3 can be matched. See Example 57 on p. 47 of the Burstein-Shapiro reference. - Alexander Burstein, Jun 03 2022

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 366*x^5 + 2148*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 167*x^4 + 942*x^5 + 5593*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 312*x^4 + 1794*x^5 + 10794*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 92*x^3 + 511*x^4 + 3000*x^5 + 18316*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 775*x^4 + 4651*x^5 + 28845*x^6 +...
A(x)^6 = 1 + 6*x + 33*x^2 + 188*x^3 + 1116*x^4 + 6852*x^5 + 43204*x^6 +...
where A(x) = 1 + x*(A(x)^2 - A(x)^3 + A(x)^4),
and A(x)^2 = 1 + x*(A(x)^2 + A(x)^5),
and A(x)^3 = 1 + x*(A(x)^2 + A(x)^4 + A(x)^6),
and A(x)^4 = 1 + x*(A(x)^2 + A(x)^4 + A(x)^5 + A(x)^7),
and A(x)^5 = 1 + x*(A(x)^2 + A(x)^4 + A(x)^5 + A(x)^6 + A(x)^8), etc.
The g.f. satisfies A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2) where
F(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 21*x^6 + 51*x^7 +...+ A001006(n-1)*x^n +...
is a g.f. of the Motzkin numbers (A001006, shifted right 1 place).
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 +...+ A106228(n)*x^n +...
satisfies G(x) = 1 + x*G(x)/(1 - x*G(x)^2).

Links

Crossrefs

Cf. A219534, A219535, A219536, A219538, A363573, A366400.
Cf. A001006, A106228, A271469, A364765.

Programs

  • Maple
    rec := {(36*n^4+126*n^3+126*n^2+36*n)*a(n)+(-276*n^4-1548*n^3-3198*n^2-2898*n-972)*a(n+1)+(940*n^4+7090*n^3+19916*n^2+24650*n+11316)*a(n+2)+(-845*n^4-9000*n^3-34159*n^2-53004*n-26136)*a(n+3)+(-260*n^4-5200*n^3-37454*n^2-116538*n-133128)*a(n+4)+(459*n^4+9774*n^3+77955*n^2+276012*n+366060)*a(n+5)+(-54*n^4-1242*n^3-10686*n^2-40758*n-58140)*a(n+6), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 13, a(4) = 66, a(5) = 366}:
f:= gfun:-rectoproc(rec,a(n),remember):
map(f, [$0..50]); # Robert Israel, Feb 25 2018
  • Mathematica
    nmax = 23; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x (A[x]^2 - A[x]^3 + A[x]^4)) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)
  • PARI
    /* Formula A(x) = 1 + x*(A(x)^2 - A(x)^3 + A(x)^4): */
{a(n)=local(A=1);for(i=1,n,A=1+x*(A^2-A^3+A^4) +x*O(x^n));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula using Series Reversion involving Motzkin numbers: */
{a(n)=local(A=1);A=(1+x-sqrt(1-2*x-3*x^2+x^3*O(x^n)))/(2*x); polcoeff(sqrt(1/x*serreverse(x/A^2)), n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) satisfies [from Paul D. Hanna, Mar 21 2016]: (Start)
(1) A(x)^2 = 1 + x*(A(x)^2 + A(x)^5).
(2) A(x)^3 = 1 + x*(A(x)^2 + A(x)^4 + A(x)^6).
Let F(x) = (1+x - sqrt(1 - 2*x - 3*x^2)) / (2*x), then g.f. A(x) satisfies:
(3) A(x) = sqrt( (1/x)*Series_Reversion(x/F(x)^2) ),
(4) A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2),
where F(x) = 1 + x*M(x) such that M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).
Let G(x) = 1 + x*G(x)/(1 - x*G(x)^2), then g.f. A(x) satisfies:
(5) A(x) = (1/x)*Series_Reversion(x/G(x)),
(6) A(x) = G(x*A(x)) and G(x) = A(x/G(x)).
where G(x) is the g.f. of A106228. (End)
Recurrence: 3*n*(3*n-1)*(3*n+1)*(5*n-11)*(5*n-8)*(5*n-6)*a(n) = 6*(5*n-11)*(900*n^5 - 3870*n^4 + 6033*n^3 - 4165*n^2 + 1238*n - 120)*a(n-1) - 2*(n-2)*(5*n-1)*(950*n^4 - 5510*n^3 + 11199*n^2 - 9207*n + 2430)*a(n-2) + 6*(n-3)*(n-2)*(2*n-5)*(5*n-6)*(5*n-3)*(5*n-1)*a(n-3). - Vaclav Kotesovec, Aug 19 2013
a(n) ~ sqrt(300+75*10^(2/3)+30*10^(1/3))/90 * (5/9*10^(2/3)+10/9*10^(1/3)+8/3)^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 19 2013
Recurrence: 18*n*(2*n+1)*(n+2)*(n+1)*a(n)-(6*(n+1))*(n+2)*(46*n^2+120*n+81)*a(n+1)+(2*(n+2))*(470*n^3+2605*n^2+4748*n+2829)*a(n+2)-(n+3)*(845*n^3+6465*n^2+14764*n+8712)*a(n+3)-(2*(n+4))*(130*n^3+2080*n^2+10407*n+16641)*a(n+4)+(3*(n+5))*(153*n^3+2493*n^2+13520*n+24404)*a(n+5)-(6*(n+5))*(3*n+17)*(3*n+19)*(n+6)*a(n+6) = 0. - Robert Israel, Feb 25 2018
G.f. A(x) satisfies: A(-x*A(x)^5) = 1/A(x). - Alexander Burstein, Jun 03 2022
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+k,n-1-k) for n > 0. - Seiichi Manyama, Aug 05 2023
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-2*k,n-1-k) for n > 0. - Seiichi Manyama, Aug 06 2023
G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A366400. - Seiichi Manyama, Mar 31 2024
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+3*k/2+1/2,n)/(2*n+3*k+1). - Seiichi Manyama, Apr 04 2024
G.f.: Sum_{k>=0} binomial(5*k/2 + 1/2, k)*x^k/((5*k + 1)*(1 - x)^((5*k + 1)/2)). - Miles Wilson, Feb 02 2025

A219534 G.f. satisfies A(x) = 1 + x*(A(x)^2 + A(x)^4).

Original entry on oeis.org

1, 2, 12, 100, 968, 10208, 113792, 1318832, 15732064, 191878592, 2381917824, 29995598208, 382257383168, 4920505410816, 63882881030656, 835554927932160, 10999486798112256, 145626782310460416, 1937772463214168064, 25901381584638605312, 347618773649248088064
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 968*x^4 + 10208*x^5 +...
Related expansions:
A(x)^2 = 1 + 4*x + 28*x^2 + 248*x^3 + 2480*x^4 + 26688*x^5 +...
A(x)^4 = 1 + 8*x + 72*x^2 + 720*x^3 + 7728*x^4 + 87104*x^5 +...
The g.f. satisfies A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) where
G(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 40*x^4 + 144*x^5 + 544*x^6 +...+ A025227(n+1)*x^n +...

Links

Crossrefs

Cf. A025227, A219535, A219536, A219537, A219538.
Cf. A006318, A364167.

Programs

  • Mathematica
    nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[1+x*(AGF^2+AGF^4)-AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *)
  • PARI
    /* Formula A(x) = 1 + x*(A(x)^2 + A(x)^4): */
{a(n)=local(A=1);for(i=1,n,A=1+x*(A^2+A^4) +x*O(x^n));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula using Series Reversion: */
{a(n)=local(A=1,G=(1-sqrt(1-4*x-4*x^2+x^3*O(x^n)))/(2*x));A=sqrt((1/x)*serreverse(x/G^2));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

Let G(x) = (1 - sqrt(1-4*x-4*x^2))/(2*x), then g.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion(x/G(x)^2) ),
(2) A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2),
where x*G(x) is the g.f. of A025227.
Recurrence: 3*n*(3*n-1)*(3*n+1)*(131*n^3 - 666*n^2 + 1075*n - 558)*a(n) = 2*(26200*n^6 - 172500*n^5 + 431572*n^4 - 521613*n^3 + 316327*n^2 - 89058*n + 8640)*a(n-1) - 12*(n-2)*(1441*n^5 - 8767*n^4 + 19186*n^3 - 18172*n^2 + 6930*n - 810)*a(n-2) + 8*(n-3)*(n-2)*(2*n-5)*(131*n^3 - 273*n^2 + 136*n - 18)*a(n-3). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 2/81*(7217783 + 10611 * sqrt(786))^(1/3) + 74654/(81*(7217783 + 10611 * sqrt(786))^(1/3)) + 400/81 = 14.48001092254652246... is the root of the equation -16 + 132*d - 400*d^2 + 27*d^3 = 0 and c = 1/2358*sqrt(262)*sqrt((213070976 + 3034746 * sqrt(786))^(1/3) * ((213070976 + 3034746 * sqrt(786))^(2/3) + 336670 + 1310*(213070976 + 3034746 * sqrt(786))^(1/3)))/((213070976 + 3034746 * sqrt(786))^(1/3)*sqrt(Pi)) = 0.1929450901182412149... - Vaclav Kotesovec, Sep 10 2013
a(n) = (1/n) * Sum_{k=0..floor(n-1)/2} 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k+1,n)/(2*n+2*k+1). - Seiichi Manyama, Apr 03 2024

A100327 Row sums of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324.

Original entry on oeis.org

1, 2, 8, 42, 252, 1636, 11188, 79386, 579020, 4314300, 32697920, 251284292, 1953579240, 15336931928, 121416356108, 968187827834, 7769449728780, 62696580696172, 508451657412496, 4141712433518956, 33872033298518728, 278014853384816184, 2289376313410678312
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2004

Keywords

Comments

Self-convolution yields A100328, which equals column 1 of triangle A100326 (omitting leading zero).

Links

Crossrefs

Cf. A003168, A003169, A100326, A219538.

Programs

  • Magma
    A100327:= func< n | n eq 0 select 1 else (2/n)*(&+[Binomial(n, k)*Binomial(2*n+k, k-1): k in [1..n]]) >;
[A100327(n): n in [0..30]]; // G. C. Greubel, Jan 30 2023
  • Maple
    A100327 := n -> simplify(2^n*binomial(3*n,2*n)*hypergeom([-1-2*n,-n], [-3*n], 1/2)/ (n+1/2)): seq(A100327(n), n=0..22); # Peter Luschny, Jun 10 2017
  • Mathematica
    Flatten[{1,Table[Sum[2*Binomial[n,k]*Binomial[2n+k,k-1]/n,{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 17 2012 *)
  • PARI
    a(n)=if(n==0,1,sum(k=0,n,2*binomial(n,k)*binomial(2*n+k,k-1)/n))
  • PARI
    a(n)=polcoeff((1/x)*serreverse(x*(1-x+sqrt(1-4*x +x^2*O(x^n)))/(2+x)),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 22 2012
  • SageMath
    def A100327(n): return 2^n*binomial(3*n,2*n)*simplify(hypergeometric([-1-2*n,-n], [-3*n],1/2)/(n+1/2))
[A100327(n) for n in range(31)] # G. C. Greubel, Jan 30 2023

Formula

G.f.: (1/x)*Series_Reversion( x*(1-x + sqrt(1 - 4*x)) / (2+x) ). - Paul D. Hanna, Nov 22 2012
G.f. A(x) = (1+G(x))/(1-G(x)), also A(x)^2 = (1+G(x))*G(x)/x, where G(x) = x*(1+G(x))/(1-G(x))^2 is the g.f. of A003169.
a(n) = 2*A003168(n) for n>0 with a(0)=1.
a(n) = Sum_{k=1..n} 2*binomial(n, k)*binomial(2n+k, k-1)/n for n>0 with a(0)=1.
Recurrence: 20*n*(2*n+1)*a(n) = (371*n^2 - 395*n + 96)*a(n-1) - 6*(27*n^2 - 103*n + 96)*a(n-2) + 4*(n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(4046 + 1122*sqrt(17))*((71 + 17*sqrt(17))/16)^n/(136*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
a(n) = 2^n*binomial(3*n,2*n)*hypergeometric([-1-2*n,-n], [-3*n],1/2)/(n+1/2). - Peter Luschny, Jun 10 2017

A371658 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x))^2.

Original entry on oeis.org

1, 4, 48, 784, 14784, 302976, 6555648, 147380480, 3408817152, 80592320512, 1938923790336, 47314993324032, 1168315059240960, 29136848453632000, 732857340425011200, 18569095605771632640, 473534596510970019840, 12144227894941523116032
Offset: 0

Views

Author

Seiichi Manyama, Apr 01 2024

Keywords

Links

Crossrefs

Cf. A219534, A219537, A371657.
Cf. A025225, A371655, A371661.
Cf. A219538.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 4^(n-k)*binomial(n, k)*binomial(3*n-k, n-1-2*k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 4^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0.
a(n) = 2^n * A219538(n). - Seiichi Manyama, Dec 26 2024

A371406 Expansion of (1/x) * Series_Reversion( x / ( (1+x)^2 * (1+2*x)^2 ) ).

Original entry on oeis.org

1, 6, 49, 462, 4734, 51216, 575705, 6657846, 78703438, 946740132, 11551512042, 142616584380, 1778372098000, 22365031140900, 283341912929865, 3612782260978470, 46326552943960278, 597034029166804068, 7728885814331709374, 100458438481544424996
Offset: 0

Views

Author

Seiichi Manyama, Mar 21 2024

Keywords

Links

Crossrefs

Cf. A219538, A371398.
Cf. A379546, A379547.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+2*x)^2))/x)
  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(2*(n+1), k)*binomial(2*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^k * binomial(2*(n+1),k) * binomial(2*(n+1),n-k).
a(n) = A219538(n+1)/2. - Seiichi Manyama, Dec 24 2024
a(n) = (1/(n+1)) * [x^n] ( (1+x) * (1+2*x) )^(2*(n+1)). - Seiichi Manyama, Dec 25 2024

A371669 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x))^2/2.

Original entry on oeis.org

1, 2, 16, 178, 2300, 32380, 481932, 7458370, 118809868, 1935217180, 32083715344, 539615356884, 9184652815816, 157908543871712, 2738272978314500, 47837620415491554, 841151610003847564, 14874918252400486060, 264381545177102073600, 4720297172922980155740
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Crossrefs

Cf. A068875, A100327, A219538.
Cf. A371661.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 2^(n-2*k)*binomial(n, k)*binomial(4*n-k, n-1-2*k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-2*k) * binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0.
Showing 1-6 of 6 results.

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