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 1-2 of 2 results.

A182304 G.f. satisfies: A(x) = x + d/dx A(x)^3/3.

Original entry on oeis.org

1, 1, 4, 25, 200, 1890, 20248, 240069, 3102000, 43226590, 644778520, 10237504810, 172274940656, 3061868388100, 57310884482160, 1126953452418765, 23230411494361920, 501015558669312390, 11285603304699928600, 265077311648541268030, 6482435029801210815280
Offset: 1

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Author

Paul D. Hanna, Apr 23 2012

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 200*x^5 + 1890*x^6 + 20248*x^7 +...
Related expansions.
A'(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 1000*x^4 + 11340*x^5 + 141736*x^6 +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 58*x^5 + 466*x^6 + 4380*x^7 + 46501*x^8 +...
A(x)^3 = x^3 + 3*x^4 + 15*x^5 + 100*x^6 + 810*x^7 + 7593*x^8 + 80023*x^9 +...
d/dx A(x)^3 = 3*x^2 + 12*x^3 + 75*x^4 + 600*x^5 + 5670*x^6 + 60744*x^7 +...

Links

Crossrefs

Cf. A245118, A088716.

Programs

  • PARI
    {a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=x+deriv(A^3/3+O(x^31)));polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    /* From g.f. F(x) of A245118 (Paul D. Hanna, Jul 27 2014): */
{a(n)=local(F=1+x); for(i=1, n, F = 1 + x*F^3/(F - x*F' +x*O(x^n))); polcoeff(serreverse(x/F), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = x + A'(x)*A(x)^2.
a(n) ~ c * n! * n^3, where c = 0.0147556981601927885048672335828437... - Vaclav Kotesovec, Feb 22 2014
G.f.: A(x) = Series_Reversion(x/F(x)) where F(x) = 1 + x*F(x)^3/(F(x) - x*F'(x)) is the g.f. of A245118. - Paul D. Hanna, Jul 27 2014

A245768 G.f. satisfies: A(x) = 1 + x*A(x)^4 / (A(x) - x*A'(x)).

Original entry on oeis.org

1, 1, 4, 26, 224, 2337, 28088, 377144, 5544824, 88039724, 1494960308, 26954440490, 513267546824, 10279486681982, 215822203235952, 4737785187211908, 108509135362455192, 2588049036893027820, 64180886929824389840, 1652564046132761428040, 44124859215715377422552, 1220338620776444854394561
Offset: 0

Views

Author

Paul D. Hanna, Aug 01 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 224*x^4 + 2337*x^5 + 28088*x^6 +...
The table of coefficients of x^k in A(x)^n begin:
n=1: [1,  1,  4,  26,  224,  2337,  28088,   377144, ...];
n=2: [1,  2,  9,  60,  516,  5330,  63318,   840808, ...];
n=3: [1,  3, 15, 103,  888,  9105, 107050,  1406655, ...];
n=4: [1,  4, 22, 156, 1353, 13804, 160844,  2092748, ...];
n=5: [1,  5, 30, 220, 1925, 19586, 226480,  2919840, ...];
n=6: [1,  6, 39, 296, 2619, 26628, 305979,  3911688, ...];
n=7: [1,  7, 49, 385, 3451, 35126, 401625,  5095392, ...];
n=8: [1,  8, 60, 488, 4438, 45296, 515988,  6501760, ...];
n=9: [1,  9, 72, 606, 5598, 57375, 651948,  8165700, ...];
n=10:[1, 10, 85, 740, 6950, 71622, 812720, 10126640, ...]; ...
in which the diagonals illustrate the relation
[x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+3) for n>=1
as follows:
[x^1] A(x)^2 = 2 = 2*[x^0] A(x)^4 = 2*1 ;
[x^2] A(x)^3 = 15 = 3*[x^1] A(x)^5 = 3*5 ;
[x^3] A(x)^4 = 156 = 4*[x^2] A(x)^6 = 4*39 ;
[x^4] A(x)^5 = 1925 = 5*[x^3] A(x)^7 = 5*385 ;
[x^5] A(x)^6 = 26628 = 6*[x^4] A(x)^8 = 6*4438 ;
[x^6] A(x)^7 = 401625 = 7*[x^5] A(x)^9 = 7*57375 ;
[x^7] A(x)^8 = 6501760 = 8*[x^6] A(x)^10 = 8*812720 ; ...
Also, from the above table, we can generate:
[1/1, 2/2, 15/3, 156/4, 1925/5, 26628/6, 401625/7, 812720/8, ...]
= [1, 1, 5, 39, 385, 4438, 57375, 812720, 12428977, 203183595, ...];
the g.f. G(x) of which begins:
G(x) = x + x^2 + 5*x^3 + 39*x^4 + 385*x^5 + 4438*x^6 + 57375*x^7 +...
such that:
G(x) = x*G(x)^4 + x^2*G(x)^3*G'(x) and G(x) = A(G(x)).

Crossrefs

Cf. A088715, A245118.

Programs

  • PARI
    /* From [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+3): */
{a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A]=((#A)*Vec(Ser(A)^(#A+2))[#A-1]-Vec(Ser(A)^(#A))[#A])/(#A)); A[n+1]}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From A(x) = 1 +  x*A(x)^4 / (A(x) - x*A'(x)): */
{a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A^4/(A - x*A' +x*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From A(x) = x/Series_Reversion(G) where G = x*G^4 + x^2*G^3*G': */
{a(n)=local(G=1+x); for(i=1, n, G = 1 + x*G^4 + x^2*G^3*G' +x*O(x^n)); polcoeff(x/serreverse(x*G +x^2*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+3) for n>=1.
(2) A(x) = x/Series_Reversion(G(x)) where G(x) = x*G(x)^4 + x^2*G(x)^3*G'(x).
Showing 1-2 of 2 results.

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