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 1-10 of 21 results. Next

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

A271469 G.f. satisfies A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5).

Original entry on oeis.org

1, 1, 4, 23, 155, 1142, 8910, 72350, 605056, 5175866, 45077560, 398348733, 3562916317, 32192775763, 293410452560, 2694283228653, 24902681767987, 231496130358758, 2162985033344112, 20301976721356134, 191336242071696514, 1809916398759630481, 17178063381786563194, 163536967014934201972, 1561247114394683682834, 14943175106109268856975
Offset: 0

Views

Author

Paul D. Hanna, Apr 08 2016

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 155*x^4 + 1142*x^5 + 8910*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 54*x^3 + 372*x^4 + 2778*x^5 + 21873*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 94*x^3 + 663*x^4 + 5025*x^5 + 39970*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 144*x^3 + 1041*x^4 + 8016*x^5 + 64470*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 205*x^3 + 1520*x^4 + 11901*x^5 + 96850*x^6 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 278*x^3 + 2115*x^4 + 16848*x^5 + 138816*x^6 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 364*x^3 + 2842*x^4 + 23044*x^5 + 192325*x^6 +...
where A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5),
and A(x)^2 = 1 + x*(A(x)^3 + A(x)^6),
and A(x)^3 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^7),
and A(x)^4 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^6 + A(x)^8),
and A(x)^5 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^9), etc.
The g.f. satisfies A(x) = F(x*A(x)^3) and F(x) = A(x/F(x)^3) 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 + 3*x^2 + 13*x^3 + 66*x^4 + 366*x^5 + 2148*x^6 +...+ A219537(n)*x^n +...
satisfies G(x) = 1 + x*(G(x)^2 - G(x)^3 + G(x)^4).

Links

Crossrefs

Cf. A363573, A001006, A370474.
Cf. A001006, A106228, A219537, A364765, A365194.
Cf. A001764, A370476.

Programs

  • Mathematica
    CoefficientList[(1/x*InverseSeries[Series[8*x^4/(1 + x - Sqrt[1 - 2*x - 3*x^2])^3, {x, 0, 20}], x])^(1/3), x] (* Vaclav Kotesovec, Apr 16 2016 *)
  • PARI
    /* Formula A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5): */
{a(n)=local(A=1); for(i=1, n, A=1+x*(A^3-A^4+A^5) +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( (1/x*serreverse(x/A^3))^(1/3), n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x)^2 = 1 + x*(A(x)^3 + A(x)^6).
(2) A(x)^3 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^7).
Let F(x) = (1+x - sqrt(1 - 2*x - 3*x^2)) / (2*x), then g.f. A(x) satisfies:
(3) A(x) = ( (1/x)*Series_Reversion(x/F(x)^3) )^(1/3),
(4) A(x) = F(x*A(x)^3) and F(x) = A(x/F(x)^3),
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)^2 - G(x)^3 + G(x)^4), 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 A219537.
a(n) ~ sqrt((34 + (34102 - 8262*sqrt(17))^(1/3) + (34102 + 8262*sqrt(17))^(1/3)) / 1632) * ((28 + (513243 - 4131*sqrt(17))^(1/3)/3 + (19009 + 153*sqrt(17))^(1/3)) / 8)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Apr 16 2016
D-finite recurrence: 8*n*(2*n-1)*(4*n-1)*(4*n+1)*(204*n^4 - 1341*n^3 + 3191*n^2 - 3286*n + 1242)*a(n) = 12*(45696*n^8 - 391776*n^7 + 1376164*n^6 - 2580579*n^5 + 2808064*n^4 - 1797694*n^3 + 651566*n^2 - 119476*n + 8160)*a(n-1) - 6*(n-2)*(29376*n^7 - 237168*n^6 + 760044*n^5 - 1236774*n^4 + 1082233*n^3 - 496791*n^2 + 108530*n - 8400)*a(n-2) + 9*(n-3)*(n-2)*(3*n-8)*(3*n-4)*(204*n^4 - 525*n^3 + 392*n^2 - 111*n + 10)*a(n-3). - Vaclav Kotesovec, Apr 16 2016
From Seiichi Manyama, Aug 06 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(3*n+k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(5*n-2*k,n-1-k) for n > 0. (End)
G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A370474. - Seiichi Manyama, Mar 31 2024
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n/2+3*k/2+1/2,n)/(3*n+3*k+1). - Seiichi Manyama, Apr 04 2024

A366400 G.f. A(x) satisfies A(x) = (1 + x * A(x)^(5/2)) / (1 - x).

Original entry on oeis.org

1, 2, 7, 32, 167, 942, 5593, 34438, 217888, 1407938, 9252168, 61641846, 415412036, 2826736736, 19395080061, 134034296976, 932110471089, 6518146460274, 45805553781349, 323313555424924, 2291130483593189, 16294149468133930, 116259325138469680
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A349311, A366401, A366402, A366403, A366404, A366405, A366406, A366407.
Cf. A262441, A370474.
Cf. A219537, A378890.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k/2, n-k)*binomial(5*k/2, k)/(3*k/2+1));
  • PARI
    a(n, r=2, s=-1, t=4, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 12 2024

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k/2,n-k) * binomial(5*k/2,k) / (3*k/2+1).
From Seiichi Manyama, Dec 12 2024: (Start)
G.f. A(x) satisfies:
(1) A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)) )^2.
(2) A(x) = 1/( 1 - x*A(x)^(3/2)/(1 + x*A(x)) )^2.
(3) A(x) = 1 + x * A(x) * (1 + A(x)^(3/2)).
(4) A(x) = B(x)^2 where B(x) is the g.f. of A219537.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)
G.f.: Sum_{k>=0} binomial(5*k/2, k)*x^k/((3*k/2 + 1)*(1 - x)^(5*k/2 + 1)). - Miles Wilson, Feb 02 2025

A300048 G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^2 + x*A(x)^6.

Original entry on oeis.org

1, 1, 2, 7, 29, 131, 627, 3124, 16032, 84162, 449828, 2439550, 13391105, 74256824, 415357737, 2340775363, 13278009018, 75753246286, 434392031856, 2502289328542, 14473290097526, 84023214062635, 489424396591995, 2859551104564120, 16754209625090980, 98415932763515679, 579475837597933632, 3419452319373566239, 20219028961691299994
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2018

Keywords

Comments

Compare to: G(x)^3 = 1 + x*G(x) + x*G(x)^2 + x*G(x)^3 holds when G(x) = 1/(1-x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 131*x^5 + 627*x^6 + 3124*x^7 + 16032*x^8 + 84162*x^9 + 449828*x^10 + 2439550*x^11 + 13391105*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 348*x^5 + 1681*x^6 + 8432*x^7 + 43495*x^8 + 229260*x^9 + 1229371*x^10 + ...
A(x)^6 = 1 + 6*x + 27*x^2 + 122*x^3 + 579*x^4 + 2862*x^5 + 14588*x^6 + 76146*x^7 + 405039*x^8 + 2187756*x^9 + 11967426*x^10 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 684*x^5 + 3341*x^6 + 16896*x^7 + 87702*x^8 + 464566*x^9 + 2501178*x^10 + ...

Links

Crossrefs

Cf. A000108, A106228, A219537.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = (1 + x*A + x*A^2 + x*A^6 +x*O(x^n))^(1/3) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) ~ sqrt(1 + sqrt((9 + 40*sqrt(3))/13)) * (9 + 6*sqrt(3) + sqrt(153 + 100*sqrt(3)))^n / (sqrt(Pi) * n^(3/2) * 2^(n + 3/2) * 3^(n + 3/4)). - Vaclav Kotesovec, Aug 11 2021
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(n+2*k,n-1-k) for n > 0. - Seiichi Manyama, Aug 05 2023

A364765 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)^5).

Original entry on oeis.org

1, 1, 5, 36, 304, 2808, 27475, 279845, 2935987, 31511097, 344344868, 3818320487, 42855633210, 485923475563, 5557803724920, 64046876264292, 742908320701832, 8667090253409215, 101631581618367133, 1197190915359577973, 14160413911721178800
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2023

Keywords

Links

Crossrefs

Cf. A001006, A106228, A219537, A271469.
Cf. A349331, A364747.
Cf. A378952, A378954.

Programs

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

Formula

G.f. satisfies A(x) = 1 + x*A(x)^6 / (1 + x*A(x)^4).
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n+k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(6*n-2*k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(5*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

A219538 G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + A(x))^2/2.

Original entry on oeis.org

1, 2, 12, 98, 924, 9468, 102432, 1151410, 13315692, 157406876, 1893480264, 23103024084, 285233168760, 3556744196000, 44730062281800, 566683825859730, 7225564521956940, 92653105887920556, 1194068058333608136, 15457771628663418748, 200916876963088849992
Offset: 0

Views

Author

Paul D. Hanna, Nov 22 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 98*x^3 + 924*x^4 + 9468*x^5 + 102432*x^6 +...
Related expansions:
A(x)^2 = 1 + 4*x + 28*x^2 + 244*x^3 + 2384*x^4 + 24984*x^5 +...
A(2)^3 = 1 + 6*x + 48*x^2 + 446*x^3 + 4524*x^4 + 48588*x^5 +...
A(2)^4 = 1 + 8*x + 72*x^2 + 712*x^3 + 7504*x^4 + 82704*x^5 +...
where A(x) = 1 + x*(A(x)^2 + 2*A(x)^3 + A(x)^4)/2.
The g.f. satisfies A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2) where
F(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 28*x^4 + 84*x^5 + 264*x^6 +...+ 2*A000108(n)*x^n +...
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + 2*x + 8*x^2 + 42*x^3 + 252*x^4 + 1636*x^5 +...+ A100327(n)*x^n +...

Links

Crossrefs

Cf. A068875, A100327.
Cf. A219534, A219537, A371658.

Programs

  • Mathematica
    CoefficientList[Sqrt[1/x*InverseSeries[Series[x^3/(1-x-Sqrt[1-4*x])^2, {x, 0, 20}], x]],x] (* Vaclav Kotesovec, Dec 28 2013 *)
  • PARI
    /* Formula A(x) = 1 + x*A(x)^2*(1 + A(x))^2/2: */
{a(n)=local(A=1);for(i=1,n,A=1+x*A^2*(1+A +x*O(x^n))^2/2);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula using Series Reversion involving Catalan numbers: */
{a(n)=local(A=1);A=(1-x-sqrt(1-4*x +x^3*O(x^n)))/x; polcoeff(sqrt(1/x*serreverse(x/A^2)), n)}
for(n=0,25,print1(a(n),", "))

Formula

Let F(x) = (1-x - sqrt(1 - 4*x)) / x, then g.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion(x/F(x)^2) ),
(2) A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2),
where F(x) = 2*C(x) - 1 such that C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
Let G(x) be the g.f. of A100327, then g.f. A(x) satisfies:
(3) A(x) = (1/x)*Series_Reversion(x/G(x)),
(4) A(x) = G(x*A(x)) and G(x) = A(x/G(x)).
Recurrence: 3*n*(3*n-1)*(3*n+1)*(11*n-14)*a(n) = 3*(2*n-1)*(693*n^3 - 1575*n^2 + 1026*n - 176)*a(n-1) + 2*(n-2)*(2*n-3)*(2*n-1)*(11*n-3)*a(n-2). - Vaclav Kotesovec, Dec 28 2013
a(n) ~ sqrt(242+66*sqrt(33)) * (7+11/9*sqrt(33))^n / (66*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 28 2013
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-2*k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 02 2024

A364739 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x)^4).

Original entry on oeis.org

1, 1, 3, 14, 78, 477, 3094, 20905, 145547, 1036891, 7522335, 55382045, 412740298, 3107671807, 23604165913, 180641336755, 1391555475647, 10781886600707, 83968131035849, 656931982467460, 5160714860765430, 40692065290732340, 321937030883130021
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A001003, A001764, A219537, A364740.

Programs

  • Mathematica
    nmax = 22; A[_] = 1;
Do[A[x_] = 1 + x*A[x]^2/(1-x*A[x]^4) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)
  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(2*n+2*k, n-1-k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k,n-1-k) for n > 0.
From Paul D. Hanna, Aug 19 2023: (Start)
A(x)^m = Sum_{n>=0} a(n,m) * x^n, where a(n,m) = (m/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k+m-1,n-1-k) for n > 0.
A(x) = exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k-1,n-1-k) for n >= 1.
G.f. A(x) satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 - A(x)^4 + A(x)^5).
(2) A(x)^2 = 1 + x*(A(x)^2 + A(x)^3 - A(x)^4 + A(x)^6).
(3) A(x)^3 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^7).
(4) A(x)^4 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^8).
(5) A(x)^5 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^6 + A(x)^9).
(6) A(x)^6 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^10).
(End)

A364758 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 + x*A(x)).

Original entry on oeis.org

1, 1, 3, 14, 76, 450, 2818, 18352, 123028, 843345, 5884227, 41650479, 298352365, 2158751879, 15754446893, 115830820439, 857147952469, 6379136387303, 47715901304501, 358529599468636, 2704884469806606, 20481615947325089, 155605509972859999, 1185779099027494848
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A090192, A106228, A364759, A378919.
Cf. A001764, A219537, A364865, A365224.
Cf. A300048, A364747, A378889.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);
  • PARI
    a(n, r=1, s=-1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 11 2024

Formula

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 11 2024: (Start)
G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^5 + x*A(x)^6.
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 + x*A(x))).
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

A363982 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 1, -1, -8, -16, 16, 195, 491, -317, -6293, -18608, 4610, 230385, 780625, 107615, -9028280, -34695607, -17401607, 367885509, 1598468196, 1350497961, -15317000747, -75391402496, -88375867528, 643505487144, 3611942077216, 5376931884971
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A291534, A364051, A364764.
Cf. A219537.

Programs

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

Formula

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

A365194 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^6).

Original entry on oeis.org

1, 1, 6, 52, 529, 5889, 69462, 853013, 10791018, 139659604, 1840435530, 24611295075, 333132371248, 4555465710569, 62839303262352, 873363902976309, 12218178082489873, 171918448407833112, 2431415226089290680, 34544425914499450493, 492807213597429920649
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A243667, A349332, A364748, A365192, A365193.
Cf. A219537, A271469, A364765.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(6*n-k+1, k)*binomial(n-1, n-k)/(6*n-k+1));

Formula

a(n) = Sum_{k=0..n} binomial(6*n-k+1,k) * binomial(n-1,n-k)/(6*n-k+1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5*n+2*k+1,k) * binomial(n-1,n-k)/(5*n+2*k+1).
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(6*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Dec 26 2024
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