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 1-4 of 4 results.

A215623 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^4).

Original entry on oeis.org

1, 2, 11, 89, 836, 8551, 92445, 1039030, 12019135, 142151324, 1711116646, 20894534324, 258195565959, 3222677162409, 40569811695707, 514520507077695, 6567611974106756, 84310605465652750, 1087798325715407703, 14098475168420865396, 183465816241394787196
Offset: 0

Views

Author

Paul D. Hanna, Aug 17 2012

Keywords

Comments

The radius of convergence of g.f. A(x) is r = 0.0712256396327314729661274986100... with A(r) = 1.4248895273944523042559975726479124492235978714420... where y=A(r) satisfies 3*y^7 - 4*y^6 + 16*y^5 - 28*y^4 + 8*y^3 - 4 = 0.

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + 8551*x^5 + 92445*x^6 + ...
Related expansions.
A(x)^4 = 1 + 8*x + 68*x^2 + 652*x^3 + 6750*x^4 + 73544*x^5 + 831078*x^6 + ...
A(x)^5 = 1 + 10*x + 95*x^2 + 965*x^3 + 10350*x^4 + 115507*x^5 + ...
where A(x) = 1 + x*(A(x) + A(x)^4) + x^2*A(x)^5.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^3)*x + (1 + 2^2*A(x)^3 + A(x)^6)*x^2/2 +
  (1 + 3^2*A(x)^3 + 3^2*A(x)^6 + A(x)^9)*x^3/3 +
  (1 + 4^2*A(x)^3 + 6^2*A(x)^6 + 4^2*A(x)^9 + A(x)^12)*x^4/4 +
  (1 + 5^2*A(x)^3 + 10^2*A(x)^6 + 10^2*A(x)^9 + 5^2*A(x)^12 + A(x)^15)*x^5/5 + ...
more explicitly,
log(A(x)) = 2*x + 18*x^2/2 + 209*x^3/3 + 2550*x^4/4 + 32082*x^5/5 + 411705*x^6/6 + 5356416*x^7/7 + ....

Links

Crossrefs

Cf. A198953, A215624, A036765, A198951, A181734, A364331, A364336, A364376.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^(3*j))*x^m/m))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x*A)*(1+x*A^4)+x*O(x^n)); polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

Formula

G.f. satisfies A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(3*k)).
The formal inverse of g.f. A(x) is (sqrt((1-x^3)^2 + 4*x^4) - (1+x^3))/(2*x^4).
a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(n+3*k+1,n-k) / (n+3*k+1). - Seiichi Manyama, Jul 19 2023
From Peter Bala, Sep 10 2024: (Start)
x/series_reversion(x*A(x)) = 1 + 2*x + 7*x^2 + 39*x^3 + 242*x^4 + 1634*x^5 + ..., the g.f. of A364336.
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 15*x^2 + 163*x^3 + 2070*x^4 + 28698*x^5 + ..., the g.f. of A364331. (End)

A364374 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^2).

Original entry on oeis.org

1, 0, -1, 1, 2, -6, -1, 28, -31, -98, 288, 131, -1730, 1638, 7431, -19583, -15502, 135642, -99523, -664050, 1535896, 1816196, -11902728, 5944326, 64487669, -129346490, -213116764, 1112382523, -277762230, -6572175490, 11287106695, 25078981772, -107983368519, -1826241850
Offset: 0

Views

Author

Seiichi Manyama, Jul 21 2023

Keywords

Links

Crossrefs

Cf. A364375, A364376.
Cf. A057078, A007863, A364371.

Programs

  • Maple
    A364374 := proc(n)
    add( (-1)^k*binomial(n+k+1,k) * binomial(n+k+1,n-k)/(n+k+1),k=0..n) ;
end proc:
seq(A364374(n),n=0..80); # R. J. Mathar, Jul 25 2023
  • Mathematica
    nmax = 33;
A[_] = 1;
Do[A[x_] = (1+x*A[x])*(1-x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 21 2023 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+k+1, k)*binomial(n+k+1, n-k)/(n+k+1));

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1).
D-finite with recurrence 15*n*(n+1)*a(n) +2*n*(13*n-11)*a(n-1) +12*(9*n^2-19*n+9)*a(n-2) +2*(10*n^2-65*n+99)*a(n-3) -4*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 25 2023
A(x) = (1/x) * series_reversion(x*(1 + x + x^2)/(1 + x)). - Peter Bala, Sep 08 2024

A364372 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^3).

Original entry on oeis.org

1, 0, -1, 3, -6, 6, 15, -107, 349, -672, 39, 5835, -27654, 75765, -95799, -279129, 2297970, -8377854, 17663640, -996624, -177445221, 888491025, -2551959604, 3337931168, 10407149226, -87719805853, 328682535695, -708428979213, 15252552804, 7616368090377, -38693979668535
Offset: 0

Views

Author

Seiichi Manyama, Jul 20 2023

Keywords

Crossrefs

Cf. A364371, A364373, A364376.
Cf. A364336.

Programs

  • Maple
    A364372 := proc(n)
    add( (-1)^k*binomial(3*k+1,k) * binomial(3*k+1,n-k)/(3*k+1),k=0..n) ;
end proc:
seq(A364372(n),n=0..80); # R. J. Mathar, Jul 25 2023
  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(3*k+1, k)*binomial(3*k+1, n-k)/(3*k+1));

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).
D-finite with recurrence 2*n*(2*n+1)*a(n) +(51*n-26)*(n-1)*a(n-1) +(279*n^2 -931*n +766)*a(n-2) +2*(413*n^2 -2127*n +2728)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 25 2024: (Start)
Fifth-order recurrence: 2*(n-2)*(n-1)*n*(2*n+1)*a(n) + (n-2)*(n-1)*(31*n^2-27*n+6)*a(n-1) + 3*(n-2)*(3*n-5)*(12*n^2-19*n+2)*a(n-2) + 3*(3*n-8)*(18*n^3-69*n^2+63*n-2)*a(n-3) + 3*n*(3*n-11)*(12*n^2-49*n+42)*a(n-4) + 3*n*(n-1)*(3*n-10)*(3*n-14)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 3 and a(4) = -6.
The g.f. A(x) satisfies (1/x) * series_reversion(x/A(x)) = 1 - x^2 + 3*x^3 - 4*x^4 - 9*x^5 + 73*x^6 - ..., the g.f. of A364376. (End)

A364375 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^3).

Original entry on oeis.org

1, 0, -1, 2, 0, -11, 28, 1, -206, 564, 38, -4711, 13329, 1273, -119762, 344707, 41884, -3251250, 9445976, 1381154, -92305098, 269504686, 45848871, -2707126108, 7921304973, 1532928960, -81375728566, 238196143730, 51591751698, -2493907008116, 7293147604136
Offset: 0

Views

Author

Seiichi Manyama, Jul 21 2023

Keywords

Crossrefs

Cf. A364371, A364372, A364374, A364376.
Cf. A198953.

Programs

  • Maple
    A364375 := proc(n)
    add( (-1)^k*binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k)/(n+2*k+1),k=0..n) ;
end proc:
seq(A364375(n),n=0..80); # R. J. Mathar, Jul 25 2023
  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+2*k+1, k)*binomial(n+2*k+1, n-k)/(n+2*k+1));

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1).
D-finite with recurrence +2*n*(191553133*n -462036810)*(2*n+1) *(n+1)*a(n) +2*n*(6735679202*n^3 -31340869996*n^2 +39568451245*n -13340358389)*a(n-1) +6*(13937077342*n^4 -106287464449*n^3 +278022830194*n^2 -296712736455*n +108876423952)*a(n-2) +6*(42118990776*n^4 -422141236704*n^3 +1546534911485*n^2 -2448212978721*n +1409411956166)*a(n-3) +6*(72631772298*n^4 -948761263665*n^3 +4512788370945*n^2 -9254886913710*n +6888712179986)*a(n-4) +3*(10147840245*n^4 -513806508936*n^3 +5519825354705*n^2 -22028093493130*n +30003008863784)*a(n-5) +6*(-9503341830*n^4 +235269814455*n^3 -2064338754902*n^2 +7709425316943*n -10409244067330)*a(n-6) +18*(3*n-20)*(n-6) *(156488131*n-746235854) *(3*n-13)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Aug 24 2024: (Start)
G.f. A(x) satisfies (1/x) * series_reversion(x*A(x)) = 1/G(x), where G(x) is the g.f. of A364371.
P-recursive: fifth-order recurrence: (2*n+1)*(2*n+2)*(3045*n^5-26680*n^4+84901*n^3-123566*n^2+86300*n-25368)*n*a(n) + 6*(18270*n^7-160080*n^6+500851*n^5-666969*n^4+307749*n^3+70849*n^2-76222*n+8288)*n*a(n-1) + 6*(54810*n^8-562455*n^7+2191158*n^6-3956204*n^5+2960986*n^4+88959*n^3-1045774*n^2+187688*n+69888)*a(n-2) + 6*(109620*n^8-1289340*n^7+5897421*n^6-13016841*n^5+13725877*n^4-5967199*n^3+2484230*n^2-3359528*n+1002624)*a(n-3) - 6*(54810*n^8-726885*n^7+3719313*n^6-9080919*n^5+10367473*n^4-4378276*n^3+1152956*n^2-2297912*n+768096)*a(n-4) + (3*n-7)*(3*n-12)*(3*n-14)*(3045*n^5-11455*n^4+8631*n^3+1507*n^2+2376*n-1368)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 2 and a(4) = 0. (End)
Showing 1-4 of 4 results.

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