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.

A243157 Series reversion of x*(1 - x)/(1 - x - x^3).

Original entry on oeis.org

1, 0, 0, -1, -1, -1, 2, 6, 11, 5, -21, -78, -124, -53, 335, 1096, 1727, 441, -5545, -17196, -25596, -2251, 97822, 284072, 399346, -44721, -1782873, -4876069, -6411063, 2201418, 33297536, 85893931, 104783903, -64745927, -632601621, -1541899544, -1727700472, 1642436289, 12171894474, 28062518974
Offset: 1

Views

Author

Paul D. Hanna, May 31 2014

Keywords

Comments

Compare to A243156, where the g.f. G(x) satisfies:
x = G(x) * (1 - G(x)) / (1 - G(x) - G(x)^3) such that G(0) = 1.

Examples

			G.f.: A(x) = x - x^4 - x^5 - x^6 + 2*x^7 + 6*x^8 + 11*x^9 + 5*x^10 - 21*x^11 - 78*x^12 - 124*x^13 - 53*x^14 + 335*x^15 +...
wherer A(x) = x * (1 - A(x) - A(x)^3) / (1 - A(x)).

Links

Crossrefs

Cf. A243156.

Programs

  • PARI
    {a(n)=local(A=x);A=serreverse(x*(1 - x)/(1 - x - x^3 +x*O(x^n)));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0,floor((n-1)/3),binomial(n,k)*binomial(n-2*k-2,n-1-3*k)*(-1)^k)/n} \\ Tani Akinari, May 21 2018

Formula

a(n) = (1/n)*(Sum_{k=0..floor((n-1)/3)} binomial(n,k)*binomial(n-2*k-2,n-1-3*k)*(-1)^k). - Tani Akinari, May 21 2018
D-finite with recurrence -n*(n+1)*a(n) +9*n*(n-1)*a(n-1) +6*(-6*n^2+22*n-17)*a(n-2) +27*(3*n-7)*(n-4)*a(n-3) +2*(-14*n^2+172*n-435)*a(n-4) -3*(53*n-192)*(n-5)*a(n-5) +341*(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Mar 24 2023

A367724 Expansion of g.f. A(x) satisfying x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^4) such that A(0) = 1.

Original entry on oeis.org

1, 1, 4, 18, 90, 487, 2785, 16559, 101309, 633523, 4030964, 26012632, 169842909, 1119960451, 7447798584, 49891581704, 336355963821, 2280413834539, 15538054908152, 106345134539952, 730770552189188, 5039872364942277, 34872958276988909, 242026790074533733, 1684356151039022090
Offset: 0

Views

Author

Paul D. Hanna, Nov 28 2023

Keywords

Examples

			G.f. A(x) = 1 + x + 4*x^2 + 18*x^3 + 90*x^4 + 487*x^5 + 2785*x^6 + 16559*x^7 + 101309*x^8 + 633523*x^9 + 4030964*x^10 + ...
Let R(x) = x * (1 - x) / (1 - x - x^4) then R(A(x)) = x;
however, A(R(x)) does not equal x, rather
A(R(x)) = 1 + x + 4*x^2 + 18*x^3 + 90*x^4 + 488*x^5 + 2794*x^6 + 16622*x^7 + 101732*x^8 + 636382*x^9 + 4050546*x^10 + ...

Crossrefs

Cf. A243156, A367725.

Programs

  • PARI
    {a(n)=polcoeff(x/serreverse(x*(1+serreverse(x/((1 + x)^4 + x +x*O(x^n))))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From a formula by Tani Akinari in A243156 */
{a(n)=0^n+sum(k=1, n, binomial(n, k)*binomial(4*k-n, k-1))/(n+0^n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^4).
(2) x = (1+x)*A(x) - A(x)^2 + x*A(x)^4 such that A(0) = 1.
(3) A(x) = x / Series_Reversion(x*(1 + Series_Reversion( x/((1 + x)^4 + x) ))).
(4) a(n) = (Sum_{k=1..n} binomial(n, k) * binomial(4*k-n, k-1))/n for n > 0 with a(0) = 1 (derived from a formula by Tani Akinari in A243156).

A367725 Expansion of g.f. A(x) satisfying x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^5) such that A(0) = 1.

Original entry on oeis.org

1, 1, 5, 30, 205, 1525, 12001, 98229, 827651, 7130614, 62528631, 556247554, 5007588460, 45535148222, 417625550140, 3858724742014, 35884576665516, 335616614245440, 3154800011439675, 29789198944740050, 282426795122071741, 2687467779597815314, 25658105671446219050
Offset: 0

Views

Author

Paul D. Hanna, Nov 28 2023

Keywords

Examples

			G.f. A(x) = 1 + x + 5*x^2 + 30*x^3 + 205*x^4 + 1525*x^5 + 12001*x^6 + 98229*x^7 + 827651*x^8 + 7130614*x^9 + 62528631*x^10 + ...
Let R(x) = x * (1 - x) / (1 - x - x^5) then R(A(x)) = x;
however, A(R(x)) does not equal x, rather
A(R(x)) = 1 + x + 5*x^2 + 30*x^3 + 205*x^4 + 1525*x^5 + 12002*x^6 + 98240*x^7 + 827752*x^8 + 7131535*x^9 + 62537177*x^10 + ...

Crossrefs

Cf. A243156, A367724.

Programs

  • PARI
    {a(n)=polcoeff(x/serreverse(x*(1+serreverse(x/((1 + x)^5 + x +x*O(x^n))))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From a formula by Tani Akinari in A243156 */
{a(n) = 0^n + sum(k=1, n, binomial(n, k)*binomial(5*k-n, k-1))/(n+0^n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^5).
(2) x = (1+x)*A(x) - A(x)^2 + x*A(x)^5 such that A(0) = 1.
(3) A(x) = x / Series_Reversion(x*(1 + Series_Reversion( x/((1 + x)^5 + x) ))).
(4) a(n) = (Sum_{k=1..n} binomial(n, k) * binomial(5*k-n, k-1))/n for n > 0 with a(0) = 1 (derived from a formula by Tani Akinari in A243156).

A371889 G.f. A(x) satisfies A(x) = 1 - x/A(x)^2 * (1 - A(x) - A(x)^3).

Original entry on oeis.org

1, 1, 2, 2, -1, -4, 7, 33, -5, -200, -151, 1185, 2202, -6069, -21799, 21791, 182718, 26520, -1349611, -1613331, 8674338, 21651795, -44750412, -217666394, 121538304, 1859974399, 1023915107, -13828122997, -23155237537, 86925632115, 282182920662
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2024

Keywords

Crossrefs

Cf. A243156, A307678, A364747, A365192.

Programs

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

Formula

a(n) = (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n-3*k,n-k-1) for n > 0.
Showing 1-4 of 4 results.

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