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 1-5 of 5 results.

A369617 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^3 + x) ).

Original entry on oeis.org

1, 4, 22, 146, 1079, 8525, 70468, 601816, 5268241, 47019566, 426250277, 3914020148, 36328457669, 340278596273, 3212416054283, 30534649412247, 291981031204917, 2806832429353512, 27109863184695640, 262951127248539898, 2560229132085602215
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2024

Keywords

Links

Crossrefs

Cf. A007317, A369616, A370844.
Cf. A006632, A274735.

Programs

  • Maple
    A369617 := proc(n)
    add(binomial(n+1,k) * binomial(4*n-4*k+2,n-k),k=0..n) ;
    %/(n+1) ;
end proc;
seq(A369617(n),n=0..70) ; # R. J. Mathar, Jan 28 2024
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^3+x))/x)
  • PARI
    a(n) = sum(k=0, n, binomial(n+1, k)*binomial(4*n-4*k+2, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(4*n-4*k+2,n-k).
D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-91*n^3 -32*n^2 +n+2)*a(n-1) +2*(n-1)*(465*n^2 -337*n+86)*a(n-2) -4*(n-1)*(n-2) *(219*n-187)*a(n-3) +283*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 28 2024

A274379 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x^2*A(x)^3).

Original entry on oeis.org

1, 3, 13, 70, 429, 2842, 19794, 142758, 1056655, 7980280, 61251261, 476387379, 3746317414, 29738316330, 237968639936, 1917578268288, 15546796822656, 126728260011920, 1037987924978125, 8538459191677170, 70509828893263474, 584310452973463242, 4857624566855734836, 40501472981905806550, 338594135314564168494, 2837641019938074131463, 23835438376045780734390, 200633658871150345742269, 1692132786239339256115050, 14297391426538004065333910, 121009206594941545408186768
Offset: 0

Views

Author

Paul D. Hanna, Aug 04 2016

Keywords

Examples

			 G.f.: A(x) = 1 + 3*x + 13*x^2 + 70*x^3 + 429*x^4 + 2842*x^5 + 19794*x^6 + 142758*x^7 + 1056655*x^8 + 7980280*x^9 + ...
such that A(x) = 1 + 3*x*A(x) + x^2*(3*A(x)^2 + A(x)^3) + x^3*(A(x)^3 + 3*A(x)^4) + 3*x^4*A(x)^5 + x^5*A(x)^6.

Links

Crossrefs

Cf. A181734, A274378.
Cf. A274735, A369600.

Programs

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

Formula

G.f. satisfies: A(x) = (1/x) * Series_Reversion( x*(1 - x^2*(1+x)^3) / (1+x)^3 ).
G.f. satisfies: A( x*(1 - x^2*(1+x)^3)/(1+x)^3 ) = (1+x)^3/(1 - x^2*(1+x)^3).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(3*n+3*k+3,n-2*k). - Seiichi Manyama, Jan 27 2024

A274734 G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x*A(x)^2).

Original entry on oeis.org

1, 3, 15, 94, 661, 4983, 39363, 321587, 2694860, 23035341, 200068651, 1760558682, 15663027711, 140648129383, 1273083938979, 11603500739475, 106404140837773, 980977232554344, 9087285865886766, 84541177049414342, 789545725457924023, 7399515198155161271, 69568021610270590583, 655960254857760518109, 6201585037793334756198, 58775103307105512895151
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2016

Keywords

Comments

More generally, if G(x) satisfies
G(x) = (1 + a*x*G(x))^m * (1 + b*x*G(x)^2), then
G(x) = (1/x) * Series_Reversion( x * (1 - b*x*(1 + a*x)^m) / (1 + a*x)^m ).

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 94*x^3 + 661*x^4 + 4983*x^5 + 39363*x^6 + 321587*x^7 +...

Links

Crossrefs

Cf. A181734, A274735.

Programs

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

Formula

G.f.: (1/x) * Series_Reversion( x * (1 - x*(1+x)^2) / (1+x)^2 ).
Recurrence: 31*(n-1)*n*(n+1)*(5974*n^3 - 40359*n^2 + 90115*n - 67124)*a(n) = 2*(n-1)*n*(1003632*n^4 - 7282128*n^3 + 18518502*n^2 - 18822839*n + 5649607)*a(n-1) - 2*(n-1)*(740776*n^5 - 6486068*n^4 + 21715762*n^3 - 34616651*n^2 + 26123385*n - 7413210)*a(n-2) + 2*(2*n - 5)*(65714*n^5 - 575377*n^4 + 1957337*n^3 - 3264653*n^2 + 2726129*n - 941430)*a(n-3) + 4*(n-3)*(2*n - 7)*(2*n - 5)*(5974*n^3 - 22437*n^2 + 27319*n - 11394)*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((s*(1+r*s)*(2 + s + 3*r*s^2)) / (1 + r*(1 + 6*s*(1+r*s)))) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.099424837262345547872398211374352678... and s = 2.183663565361369673488934371066403742... are roots of the system of equations (1 + r*s)^2*(1 + r*s^2) = s, 2*r*(1 + s + 2*r^2*s^3 + r*s*(1 + 3*s)) = 1. - Vaclav Kotesovec, Nov 18 2017
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(2*n+2*k+2,n-k). - Seiichi Manyama, Jan 27 2024

A369600 Expansion of (1/x) * Series_Reversion( x * (1/(1+x)^3 - x^3) ).

Original entry on oeis.org

1, 3, 12, 56, 291, 1638, 9780, 60948, 391821, 2577575, 17256918, 117150228, 804343302, 5575177026, 38957753136, 274143594685, 1941037464402, 13818185220783, 98848503602394, 710185896393792, 5122358166219855, 37076879861508830, 269235792063692580
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2024

Keywords

Links

Crossrefs

Cf. A198888, A369599.
Cf. A274379, A274735.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1/(1+x)^3-x^3))/x)
  • PARI
    a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(3*n+3*k+3, n-3*k))/(n+1);

Formula

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

A379190 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * (1 + x*A(x))^3.

Original entry on oeis.org

1, 4, 30, 304, 3557, 45150, 604222, 8393282, 119872890, 1749183075, 25964512607, 390828464403, 5951561595889, 91523131078999, 1419293428538496, 22169968253466467, 348507676062911520, 5509187208564734328, 87522347516801353980, 1396619714730284551913, 22375420057050167868366
Offset: 0

Views

Author

Seiichi Manyama, Dec 17 2024

Keywords

Crossrefs

Cf. A379172, A379188, A379191.
Cf. A002293, A274735, A364475.
Cf. A198953.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(3*n+6*k+3, n-k)/(n+2*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(3*n+6*k+3,n-k)/(n+2*k+1).
Showing 1-5 of 5 results.

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