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.

A216314 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + 2*x*A(x)^2).

Original entry on oeis.org

1, 3, 17, 121, 965, 8247, 73841, 683713, 6493145, 62898859, 619079889, 6173490857, 62239144525, 633323532783, 6496052173665, 67093423506049, 697181754821297, 7283521984427283, 76455801614169809, 806004056649062937, 8529783421905380629, 90584730265930813607
Offset: 0

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Author

Paul D. Hanna, Sep 03 2012

Keywords

Comments

The radius of convergence of g.f. A(x) is r = 0.08774268876242660659654020... with A(r) = 2.04748732367111203761312028274219344812311691... where y=A(r) satisfies 6*y^3 - 14*y^2 + 4*y - 1 = 0.
r = 1/(((40465 + 387*sqrt(129))^(2/3) + 1174 + 34*(40465 + 387*sqrt(129))^(1/3)) / (40465+387*sqrt(129))^(1/3)/9). - Vaclav Kotesovec, Sep 17 2013

Examples

			G.f.: A(x) = 1 + 3*x + 17*x^2 + 121*x^3 + 965*x^4 + 8247*x^5 + 73841*x^6 +...
Related expansions.
A(x)^2 = 1 + 6*x + 43*x^2 + 344*x^3 + 2945*x^4 + 26398*x^5 + 244615*x^6 +...
A(x)^3 = 1 + 9*x + 78*x^2 + 696*x^3 + 6399*x^4 + 60321*x^5 + 580316*x^6 +...
where A(x) = 1 + A(x)*(1+2*A(x))*x + 2*A(x)^3*x^2.
The g.f. also satisfies the series:
A(x) = 1 + 3*x*A(x) + 8*x^2*A(x)^2 + 22*x^3*A(x)^3 + 60*x^4*A(x)^4 + 164*x^5*A(x)^5 + 448*x^6*A(x)^6 +...+ A028859(n)*x^n*A(x)^n +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1*2 + 1/A(x))*x*A(x) + (1*2^2 + 2^2*2/A(x) + 1/A(x)^2)*x^2*A(x)^2/2 +
(1*2^3 + 3^2*2^2/A(x) + 3^2*2/A(x)^2 + 1/A(x)^3)*x^3*A(x)^3/3 +
(1*2^4 + 4^2*2^3/A(x) + 6^2*2^2/A(x)^2 + 4^2*2/A(x)^3 + 1/A(x)^4)*x^4*A(x)^4/4 +
(1*2^5 + 5^2*2^4/A(x) + 10^2*2^3/A(x)^2 + 10^2*2^2/A(x)^3 + 5^2*2/A(x)^4 + 1/A(x)^5)*x^5*A(x)^5/5 +...
Explicitly,
log(A(x)) = 3*x + 25*x^2/2 + 237*x^3/3 + 2361*x^4/4 + 24203*x^5/5 + 252757*x^6/6 + 2674185*x^7/7 + 28567105*x^8/8 +...+ L(n)*x^n/n +...
where L(n) = [x^n] (1+x)^n/(1-2*x-2*x^2)^n.

Links

Crossrefs

Cf. A007863, A215661, A215654, A028859, A364374.

Programs

  • Mathematica
    CoefficientList[1/x * InverseSeries[Series[x*(1 - 2*x - 2*x^2)/(1+x),{x,0,20}],x],x] (* Vaclav Kotesovec, Sep 17 2013 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + 2*x*(A+x*O(x^n))^2)); polcoeff(A, n)}
  • PARI
    {a(n)=polcoeff( (1/x)*serreverse( x*(1-2*x-2*x^2)/(1+x +x*O(x^n))), 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*2^(m-j)/A^j)*x^m*A^m/m))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * 2^(n-k)/A(x)^k ).
(2) A(x) = (1/x) * Series_Reversion( x*(1 - 2*x - 2*x^2)/(1+x) ).
(3) A(x) = Sum_{n>=0} A028859(n) * x^n * A(x)^n, where g.f. of A028859 = (1+x)/(1-2*x-2*x^2).
The formal inverse of the g.f. A(x) is (sqrt(1-4*x+12*x^2) - (1+2*x))/(4*x^2).
a(n) = [x^n] ( (1+x)/(1-2*x-2*x^2) )^(n+1) / (n+1).
Recurrence: 3*n*(n+1)*(43*n-76)*a(n) = n*(1462*n^2 - 3315*n + 1274)*a(n-1) + (86*n^3 - 324*n^2 + 523*n - 330)*a(n-2) + (n-2)*(2*n-5)*(43*n-33)*a(n-3)
a(n) ~ 1/516*sqrt(86)*sqrt((1448486261 + 1803807*sqrt(129))^(1/3)*((1448486261 + 1803807*sqrt(129))^(2/3) + 1280110 + 1118*(1448486261 + 1803807*sqrt(129))^(1/3)))/(1448486261 + 1803807*sqrt(129))^(1/3) * (((40465 + 387*sqrt(129))^(2/3) + 1174 + 34*(40465 + 387*sqrt(129) )^(1/3)) / (40465+387*sqrt(129))^(1/3)/9)^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013
a(n) = Sum_{k=0..n} 2^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024

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

A375436 Expansion of g.f. A(x) satisfying A(x) = (1 + 2*x*A(x)) * (1 + 3*x*A(x)^2).

Original entry on oeis.org

1, 5, 46, 533, 6922, 96338, 1404796, 21184229, 327659314, 5169425894, 82866843652, 1345864066658, 22098946620580, 366245357320196, 6118363978530424, 102921394554326021, 1741855452305095618, 29637960953559091934, 506708801920060974388, 8700147627314354759030, 149957787462657877848556
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 + 46*x^2 + 533*x^3 + 6922*x^4 + 96338*x^5 + 1404796*x^6 + 21184229*x^7 + 327659314*x^8 + 5169425894*x^9 + 82866843652*x^10 + ...
where A(x) = (1 + 2*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 + 5*x + 21*x^2 + 93*x^3 + 405*x^4 + 1773*x^5 + 7749*x^6 + 33885*x^7 + ... + A154964(n+1)*x^n + ...
where B(x) = (1 + 2*x)/(1 - 3*x - 6*x^2).

Links

Crossrefs

Cf. A154964, A216314, A215661, A375434, A375435, A375437.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A=(1 + 2*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 - 6*x^2)/(1 + 2*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 * 2^(m-j) * 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 the following formulas.
(1) A(x) = (1 + 2*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 * 2^(n-k) * 3^k * A(x)^k ).
(3) A(x) = (1/x) * Series_Reversion( x*(1 - 3*x - 6*x^2)/(1 + 2*x) ).
(4) A(x) = Sum_{n>=0} A154964(n+1) * x^n * A(x)^n, where g.f. of A154964 = (1 - 2*x - 4*x^2)/(1 - 3*x - 6*x^2).
(5) x = (sqrt(33*A(x)^2 - 12*A(x) + 4) - (2 + 3*A(x)))/(12*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * 2^(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
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