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 1-3 of 3 results.

A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).

Original entry on oeis.org

1, 4, 31, 301, 3274, 38158, 465919, 5883040, 76189177, 1006440238, 13508178448, 183689450959, 2525336086630, 35041483528522, 490125130328455, 6902993856515389, 97814486474787898, 1393470813699724726, 19946461692566594413, 286742046721454817358, 4138001844031453456120
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2024

Keywords

Comments

In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then
(C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).
(C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).
(C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).

Examples

			G.f. A(x) = 1 + 4*x + 31*x^2 + 301*x^3 + 3274*x^4 + 38158*x^5 + 465919*x^6 + 5883040*x^7 + 76189177*x^8 + 1006440238*x^9 + 13508178448*x^10 + ...
where A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
RELATED SERIES.
Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
B(x) = 1 + 4*x + 15*x^2 + 57*x^3 + 216*x^4 + 819*x^5 + 3105*x^6 + 11772*x^7 + ... + A125145(n)*x^n + ...
where B(x) = (1 + x)/(1 - 3*x - 3*x^2).

Links

Crossrefs

Cf. A125145, A215661, A375435, A375436, A375437.
Cf. A007863, A216314, A364374.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + 3*x*(A+x*O(x^n))^2)); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = polcoef( (1/x)*serreverse( x*(1-3*x-3*x^2)/(1+x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = my(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 * 3^j * A^j)*x^m/m))); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies:
(1) A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 3^k * A(x)^k ).
(3) A(x) = (1/x) * Series_Reversion( x*(1 - 3*x - 3*x^2)/(1 + x) ).
(4) A(x) = Sum_{n>=0} A125145(n) * x^n * A(x)^n, where g.f. of A125145 = (1 + x)/(1 - 3*x - 3*x^2).
(5) x = (sqrt(21*A(x)^2 - 6*A(x) + 1) - (1 + 3*A(x)))/(6*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024
a(n) ~ ((36 + (48266 - 714*sqrt(17))^(1/3) + (48266 + 714*sqrt(17))^(1/3))/7)^n / (sqrt(6*Pi*((20517 - 4861*sqrt(17))^(1/3) + (20517 + 4861*sqrt(17))^(1/3) - 42)) * n^(3/2)). - Vaclav Kotesovec, Sep 14 2024

A375435 Expansion of g.f. A(x) satisfying A(x) = (1 + 3*x*A(x)) * (1 + x*A(x)^2).

Original entry on oeis.org

1, 4, 23, 167, 1370, 12066, 111399, 1063896, 10423145, 104172842, 1057938416, 10886055709, 113252336950, 1189231665334, 12588038915535, 134172815937543, 1438842536532522, 15513036330871914, 168057711839246901, 1828443841807079994, 19970180509170366264, 218877585875869278396
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2024

Keywords

Comments

In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then
(C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).
(C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).
(C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).

Examples

			G.f. A(x) = 1 + 4*x + 23*x^2 + 167*x^3 + 1370*x^4 + 12066*x^5 + 111399*x^6 + 1063896*x^7 + 10423145*x^8 + 104172842*x^9 + 1057938416*x^10 + ...
where A(x) = (1 + 3*x*A(x)) * (1 + x*A(x)^2).
RELATED SERIES.
Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
B(x) = 1 + 4*x + 7*x^2 + 19*x^3 + 40*x^4 + 97*x^5 + 217*x^6 + 508*x^7 + 1159*x^8 + ... + A006130(n+1)*x^n + ...
where B(x) = (1 + 3*x)/(1 - x - 3*x^2).

Links

Crossrefs

Cf. A006130, A216314, A215661, A375434, A375436, A375437.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A=(1 + 3*x*A)*(1 + x*(A+x*O(x^n))^2)); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=polcoef( (1/x)*serreverse( x*(1 - x - 3*x^2)/(1+3*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = my(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 * 3^(m-j) * A^j)*x^m/m))); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = (1 + 3*x*A(x)) * (1 + x*A(x)^2).
(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 3^(n-k) * A(x)^k ).
(3) A(x) = (1/x) * Series_Reversion( x*(1 - x - 3*x^2)/(1 + 3*x) ).
(4) A(x) = Sum_{n>=0} A006130(n+1) * x^n * A(x)^n, where g.f. of A006130 = 1/(1 - x - 3*x^2).
(5) x = (sqrt(13*A(x)^2 - 6*A(x) + 9) - (3 + A(x)))/(6*A(x)^2).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024

A375437 Expansion of g.f. A(x) satisfying A(x) = (1 + 3*x*A(x)) * (1 + 2*x*A(x)^2).

Original entry on oeis.org

1, 5, 41, 427, 4997, 62697, 824361, 11210331, 156371609, 2224976461, 32167995497, 471208730027, 6978452945485, 104313403711649, 1571764793999769, 23847629857934859, 364033580432140593, 5586881305151655381, 86153520326218040553, 1334246446733337499755, 20743139707001572645461
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2024

Keywords

Comments

In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then
(C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).
(C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).
(C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).

Examples

			G.f. A(x) = 1 + 5*x + 41*x^2 + 427*x^3 + 4997*x^4 + 62697*x^5 + 824361*x^6 + 11210331*x^7 + 156371609*x^8 + 2224976461*x^9 + 32167995497*x^10 + ...
where A(x) = (1 + 3*x*A(x)) * (1 + 2*x*A(x)^2).
RELATED SERIES.
Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
B(x) = 1 + 5*x + 16*x^2 + 62*x^3 + 220*x^4 + 812*x^5 + 2944*x^6 + 10760*x^7 + ... + A307469(n)*x^n + ...
where B(x) = (1 + 3*x)/(1 - 3*x - 6*x^2).

Links

Crossrefs

Cf. A307469, A216314, A215661, A375434, A375435, A375436.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A=(1 + 3*x*A)*(1 + 2*x*(A+x*O(x^n))^2)); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=polcoef( (1/x)*serreverse( x*(1 - 2*x - 6*x^2)/(1 + 3*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = my(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 * 3^(m-j) * 2^j * A^j)*x^m/m))); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = (1 + 3*x*A(x)) * (1 + 2*x*A(x)^2).
(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 3^(n-k) * 2^k * A(x)^k ).
(3) A(x) = (1/x) * Series_Reversion( x*(1 - 2*x - 6*x^2)/(1 + 3*x) ).
(4) A(x) = Sum_{n>=0} A307469(n) * x^n * A(x)^n, where g.f. of A307469 = (1 + 3*x)/(1 - 3*x - 6*x^2).
(5) x = (sqrt(28*A(x)^2 - 12*A(x) + 9) - (3 + 2*A(x)))/(12*A(x)^2).
a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024
Showing 1-3 of 3 results.

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