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.

Previous Showing 21-29 of 29 results.

A366327 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^3).

Original entry on oeis.org

1, 2, -5, 33, -260, 2263, -20979, 203124, -2030121, 20786694, -216928144, 2298911699, -24673591005, 267644087524, -2929602893537, 32317666058508, -358931896710948, 4010200327457883, -45040693394259858, 508253687784232108, -5759468659295939684
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2023

Keywords

Crossrefs

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366326, A366328.
Cf. A006632, A364407.

Programs

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

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(4*k-1,k) * binomial(n+2*k-2,n-k)/(4*k-1).

A366328 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^4).

Original entry on oeis.org

1, 2, -7, 60, -612, 6898, -82806, 1038076, -13431940, 178040315, -2405137161, 32992706368, -458336721104, 6435090557964, -91167680664004, 1301665779507128, -18710805300530504, 270559054510943509, -3932893180646204203, 57437414168562779574, -842365843304975785062
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2023

Keywords

Crossrefs

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366326, A366327.
Cf. A118971, A364408.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k-1, k)*binomial(n+3*k-2, n-k)/(5*k-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k-1,k) * binomial(n+3*k-2,n-k)/(5*k-1).

A216359 G.f. satisfies: A(x) = (1 + x*A(x)^2) * (1 + x/A(x)).

Original entry on oeis.org

1, 2, 3, 13, 32, 147, 445, 2067, 7019, 32590, 119209, 551551, 2125429, 9795863, 39221165, 180177403, 742575760, 3403131833, 14342166121, 65626369612, 281459129188, 1286834885967, 5596229192396, 25580269950635, 112492633046446, 514323765191879, 2282371511598955
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2012

Keywords

Comments

The radius of convergence of g.f. A(x) is r = 0.209619875959405379599013693... with A(r) = 2.36951367232829409921688546894691317519410... where y=A(r) satisfies y^7 - 2*y^6 - 4*y^4 + 4*y^3 + 4*y - 2 = 0.

Examples

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + 445*x^6 +...
Related expansions.
A(x)^2 = 1 + 4*x + 10*x^2 + 38*x^3 + 125*x^4 + 500*x^5 + 1839*x^6 +...
A(x)^3 = 1 + 6*x + 21*x^2 + 83*x^3 + 315*x^4 + 1269*x^5 + 5061*x^6 +...
where A(x) = (1-x^2)*A(x)^2 - x*A(x)^3 - x.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + 1/A(x)^3)*x*A(x) + (1 + 2^2/A(x)^3 + 1/A(x)^6)*x^2*A(x)^2/2 +
(1 + 3^2/A(x)^3 + 3^2/A(x)^6 + 1/A(x)^9)*x^3*A(x)^3/3 +
(1 + 4^2/A(x)^3 + 6^2/A(x)^6 + 4^2/A(x)^9 + 1/A(x)^12)*x^4*A(x)^4/4 +
(1 + 5^2/A(x)^3 + 10^2/A(x)^6 + 10^2/A(x)^9 + 5^2/A(x)^12 + 1/A(x)^15)*x^5*A(x)^5/5 +...

Links

Crossrefs

Cf. A073157, A198953, A215623, A215624, A364331, A364336.

Programs

  • Maple
    S:= series(RootOf(x+y+x^2*y^2-y^2+x*y^3, y, 1), x, 41):
seq(coeff(S,x,j),j=0..40); # Robert Israel, Jul 10 2017
  • 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^2)*AGF^2 - x*AGF^3 - x - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 18 2013 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A^2)*(1 + x/(A+x*O(x^n)))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2/A^(3*j))*x^m*A^m/m))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

G.f. satisfies:
A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 / A(x)^(3*k) ).
The formal inverse of the g.f. A(x) is (sqrt(1 - 2*x^3 + 4*x^4 + x^6) - (1+x^3))/(2*x^2).
Recurrence: n*(n+1)*(1241*n^5 - 21306*n^4 + 135203*n^3 - 381522*n^2 + 435524*n - 104880)*a(n) = 6*n*(1201*n^4 - 19476*n^3 + 114613*n^2 - 287442*n + 255364)*a(n-1) + 2*(12410*n^7 - 237880*n^6 + 1771109*n^5 - 6388366*n^4 + 11032829*n^3 - 6363274*n^2 - 3856020*n + 4157712)*a(n-2) + 6*(2482*n^7 - 51299*n^6 + 419427*n^5 - 1705769*n^4 + 3477465*n^3 - 2797370*n^2 - 637684*n + 1410288)*a(n-3) + 2*(4964*n^7 - 110044*n^6 + 983093*n^5 - 4442260*n^4 + 10160177*n^3 - 8790970*n^2 - 4722180*n + 9233280)*a(n-4) - 6*(2482*n^7 - 58745*n^6 + 553921*n^5 - 2617109*n^4 + 6255337*n^3 - 6022682*n^2 - 1392300*n + 4289616)*a(n-5) + 60*(n-7)*(2*n - 11)*(n^3 - 40*n^2 + 280*n - 552)*a(n-6) + 2*(n-8)*(2*n - 13)*(1241*n^5 - 15101*n^4 + 62389*n^3 - 91339*n^2 - 930*n + 64260)*a(n-7). - Vaclav Kotesovec, Sep 18 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 4.77053998540509708... is the root of the equation -4 + 12*d^2 - 8*d^3 - 12*d^4 - 20*d^5 + d^7 = 0 and c = 1.27852844884923435863262213680985089152... - Vaclav Kotesovec, Sep 18 2013
In closed form, c = (-4 + (1 + sqrt(1+8/d^2))*d^2) * sqrt((d^3*(1 + sqrt(1+8/d^2) + (4*(4 + d^2*(-3-sqrt(1+8/d^2) + d*(4+d))))/d^6)) / (1 + 1/64*(1 + sqrt(1+8/d^2)-4/d^2)^3*d^3)) / (32*d). - Vaclav Kotesovec, Aug 18 2014
From Peter Bala, Sep 10 2024: (Start)
For n not of the form 3*m + 1, we conjecture that a(n) = Sum_{k = 0..n} binomial(-n+3*k+1, k)*binomial(-n+3*k+1, n-k)/(-n+3*k+1).
Define a sequence operator R: {u(n): n >= 0} -> {v(n): n >= 0} by Sum_{n >= 0} v(n)*x^n = (1/x) * series_reversion(x/Sum_{n >= 0} u(n)*x^n). Then R({a(n)}) = A364336, R^2({a(n)}) = A215623 and R^3({a(n)}) = A364331. Cf. A073157. (End)

A349016 G.f. A(x) satisfies: A(x) = 1 + x * A(-x) / (1 - x) + x * A(x)^2.

Original entry on oeis.org

1, 2, 3, 12, 26, 125, 317, 1642, 4492, 24188, 69174, 381613, 1123923, 6304781, 18962485, 107682542, 329007674, 1885923378, 5833166568, 33685017384, 105214504816, 611241171298, 1924588709710, 11236434464097, 35617302886643, 208815253200975, 665665428686531
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2021

Keywords

Crossrefs

Cf. A000108, A073157, A215973, A349014, A349015.

Programs

  • Mathematica
    nmax = 26; A[] = 0; Do[A[x] = 1 + x A[-x]/(1 - x) + x A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[a[k] ((-1)^k + a[n - k - 1]), {k, 0, n - 1}]; Table[a[n], {n, 0, 26}]

Formula

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

A364329 G.f. satisfies A(x) = (1 + x^3) * (1 + x*A(x)^2).

Original entry on oeis.org

1, 1, 2, 6, 17, 52, 167, 558, 1912, 6683, 23736, 85426, 310861, 1141837, 4227938, 15764474, 59140089, 223062670, 845388258, 3217750229, 12295043520, 47144444476, 181349473833, 699629022954, 2706327445312, 10494497061015, 40787775234746, 158859378070721
Offset: 0

Views

Author

Seiichi Manyama, Jul 18 2023

Keywords

Crossrefs

Cf. A073157, A215576, A364330.

Programs

  • Maple
    A364329 := proc(n)
    add( binomial(2*n-6*k+1,k) * binomial(2*n-6*k+1,n-3*k)/(2*n-6*k+1),k=0..n/3) ;
end proc:
seq(A364329(n),n=0..70); # R. J. Mathar, Jul 25 2023
  • Mathematica
    nmax = 27; A[_] = 1;
Do[A[x_] = (1 + x^3)*(1 + x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-6*k+1, k)*binomial(2*n-6*k+1, n-3*k)/(2*n-6*k+1));

Formula

G.f.: A(x) = 2*(1 + x^3) / (1 + sqrt(1-4*x*(1 + x^3)^2)).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-6*k+1,k) * binomial(2*n-6*k+1,n-3*k) / (2*n-6*k+1).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-3) +6*(-2*n+7)*a(n-4) +6*(-2*n+13)*a(n-7) +2*(-2*n+19)*a(n-10)=0. - R. J. Mathar, Jul 25 2023

A364330 G.f. satisfies A(x) = (1 + x^4) * (1 + x*A(x)^2).

Original entry on oeis.org

1, 1, 2, 5, 15, 45, 142, 464, 1556, 5327, 18532, 65326, 232826, 837589, 3037472, 11092143, 40753626, 150541422, 558762382, 2082871613, 7794301294, 29269317708, 110263451242, 416595676681, 1578183767068, 5993326380378, 22812048907856, 87010994947971, 332531385362972
Offset: 0

Views

Author

Seiichi Manyama, Jul 18 2023

Keywords

Crossrefs

Cf. A073157, A215576, A364329.

Programs

  • Maple
    A364330 := proc(n)
    add( binomial(2*n-8*k+1,k) * binomial(2*n-8*k+1,n-4*k)/(2*n-8*k+1),k=0..n/4) ;
end proc:
seq(A364330(n),n=0..80); # R. J. Mathar, Jul 25 2023
  • Mathematica
    nmax = 28; A[_] = 1;
Do[A[x_] = (1 + x^4)*(1 + x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(2*n-8*k+1, k)*binomial(2*n-8*k+1, n-4*k)/(2*n-8*k+1));

Formula

G.f.: A(x) = 2*(1 + x^4) / (1 + sqrt(1-4*x*(1 + x^4)^2)).
a(n) = Sum_{k=0..floor(n/4)} binomial(2*n-8*k+1,k) * binomial(2*n-8*k+1,n-4*k) / (2*n-8*k+1).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-4) +6*(-2*n+9)*a(n-5) +6*(-2*n+17)*a(n-9) +2*(-2*n+25)*a(n-13)=0. - R. J. Mathar, Jul 25 2023

A366236 G.f. A(x) satisfies A(x) = 1 + x + x*(1 + x)^2*A(x)^2.

Original entry on oeis.org

1, 2, 6, 25, 110, 520, 2566, 13073, 68244, 363129, 1962304, 10739914, 59411546, 331652408, 1865903040, 10569319231, 60227702736, 345015430415, 1985747398748, 11477353063881, 66590427901454, 387685469752989, 2264180109124196, 13261401158297918
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2023

Keywords

Crossrefs

Cf. A073157, A366237, A366238.
Cf. A000108.

Programs

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

Formula

G.f.: 2*(1+x) / (1 + sqrt(1-4*x*(1+x)^3)).
a(n) = Sum_{k=0..n} binomial(3*k+1,n-k) * binomial(2*k,k)/(k+1).

A366237 G.f. A(x) satisfies A(x) = 1 + x + x*(1 + x)^3*A(x)^2.

Original entry on oeis.org

1, 2, 7, 33, 161, 843, 4601, 25896, 149254, 876480, 5225616, 31547730, 192470212, 1184804588, 7349888208, 45902094845, 288368474907, 1821096958308, 11554270204142, 73615309821574, 470795634833760, 3021222108762826, 19448517295201332
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2023

Keywords

Crossrefs

Cf. A073157, A366236, A366238.
Cf. A000108.

Programs

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

Formula

G.f.: 2*(1+x) / (1 + sqrt(1-4*x*(1+x)^4)).
a(n) = Sum_{k=0..n} binomial(4*k+1,n-k) * binomial(2*k,k)/(k+1).

A366238 G.f. A(x) satisfies A(x) = 1 + x + x*(1 + x)^4*A(x)^2.

Original entry on oeis.org

1, 2, 8, 42, 224, 1281, 7630, 46816, 294008, 1880588, 12209474, 80251889, 532988530, 3571260662, 24112334292, 163887278097, 1120445503036, 7699924478714, 53160597794588, 368549236730128, 2564649436452878, 17907555498455680, 125426544531794332
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2023

Keywords

Crossrefs

Cf. A073157, A366236, A366237.
Cf. A000108.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*k+1, n-k)*binomial(2*k, k)/(k+1));

Formula

G.f.: 2*(1+x) / (1 + sqrt(1-4*x*(1+x)^5)).
a(n) = Sum_{k=0..n} binomial(5*k+1,n-k) * binomial(2*k,k)/(k+1).
Previous Showing 21-29 of 29 results.

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