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.

A153851 Nonzero coefficients of the g.f. that satisfies: A(x) = x + A(A(x))^3.

Original entry on oeis.org

1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448, 810629805, 15938815794, 326653743510, 6949638584208, 153009877730525, 3477623225388063, 81429702521625843, 1961136442605508341, 48513571089988199157
Offset: 1

Views

Author

Paul D. Hanna, Jan 21 2009

Keywords

Examples

			G.f.: A(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 +...
A(x - A(x)^3) = x where
A(x)^3 = x^3 + 3*x^5 + 21*x^7 + 208*x^9 + 2517*x^11 + 34851*x^13 +...
SYSTEM OF RELATED FUNCTIONS.
A = A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x^2*B^3;
B = A + x^2*C^3;
C = B + x^2*D^3;
D = C + x^2*E^3;
E = D + x^2*F^3; ...
where the functions xB, xC, xD, etc., are successive iterations of A(x):
x*A = A(x),
x*B = A(A(x)) = g.f. of A153852,
x*C = A(A(A(x))) = g.f. of A153853,
x*D = A(A(A(A(x)))) = g.f. of A153854, etc.
The nonzero coefficients of these functions begin:
A:[1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448,...];
B:[1, 2, 15, 165, 2213, 33693, 561867, 10053141, 190489374,...];
C:[1, 3, 27, 339, 5067, 84738, 1536867, 29687772, 603835479,...];
D:[1, 4, 42, 594, 9827, 179928, 3545637, 73988631, 1618178067,...];
E:[1, 5, 60, 945, 17180, 342765, 7316178, 164606166, 3866962617,...];
F:[1, 6, 81, 1407, 27918, 603879, 13907133, 336334443, 8466942393,...];
G:[1, 7, 105, 1995, 42938, 1001973, 24795645, 642380025, 17278647147,...];
H:[1, 8, 132, 2724, 63242, 1584768, 41975610, 1160887350, 33260962995,..]; ...
The main diagonal in the above table is A153850.

Links

Crossrefs

Cf. A153852, A153853, A153854, A153850; variants: A139702, A213591.

Programs

  • PARI
    {a(n)=local(A=x+x^2); for(i=0, n, A=serreverse(x-subst(A^3, x, x+x^2*O(x^(2*n))))) ; polcoeff(A, 2*n-1)}

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies:
(1) A(x) = Series_Reversion( x - A(x)^3 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n) / n!. - Paul D. Hanna, Sep 07 2020
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x / n! ). - Paul D. Hanna, Sep 07 2020
(4) x = A(A( x-x^3 - A(x)^3 )). - Paul D. Hanna, Sep 07 2020

A153850 a(n) is the coefficient of x^(2*n-1) in the n-fold self-composition of G(x) = x + G(G(x))^3 = g.f. of A153851.

Original entry on oeis.org

1, 2, 27, 594, 17180, 603879, 24795645, 1160887350, 60940292571, 3541938123306, 225669592036086, 15634133444509443, 1169781625911185118, 93989088711427170141, 8069678384570571946581, 737204558292074214218778
Offset: 1

Views

Author

Paul D. Hanna, Jan 21 2009

Keywords

Examples

			Let A(x) be the g.f. of A153851, which begins
A(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 + 145815*x^13 + 2430393*x^15 + 43202448*x^17 + ... + A153851(2*n-1)*x^n + ...
then A(x) satisfies A(x - A(x)^3) = x.
Further, let successive iterations of A(x) be denoted by
B(x) = A(A(x)) = g.f. of A153852,
C(x) = A(A(A(x))) = g.f. of A153853,
D(x) = A(A(A(A(x)))) = g.f. of A153854, etc.,
then the nonzero coefficients in the successive iterations of A(x) form the table:
A:[1, 1, 6, 57, 683, 9474, 145815, 2430393, ...];
B:[1, 2, 15, 165, 2213, 33693, 561867, 10053141, ...];
C:[1, 3, 27, 339, 5067, 84738, 1536867, 29687772, ...];
D:[1, 4, 42, 594, 9827, 179928, 3545637, 73988631, ...];
E:[1, 5, 60, 945, 17180, 342765, 7316178, 164606166, ...];
F:[1, 6, 81, 1407, 27918, 603879, 13907133, 336334443, ...];
G:[1, 7, 105, 1995, 42938, 1001973, 24795645, 642380025, ...];
H:[1, 8, 132, 2724, 63242, 1584768, 41975610, 1160887350, ...]; ...
in which the main diagonal equals this sequence.

Crossrefs

Cf. A153851, A153852, A153853, A153854.

Programs

  • PARI
    {a(n) = my(G=x+O(x^(2*n+1)),H=G); for(i=0, n, G=serreverse(x-G^3)); for(i=1,n,H=subst(G,x,H)); polcoeff(H, 2*n-1)}
for(n=1,20,print1(a(n),", "))

A153853 Nonzero coefficients of g.f.: A(x) = G(G(G(x))) where G(x) = x + G(G(x))^3 is the g.f. of A153851.

Original entry on oeis.org

1, 3, 27, 339, 5067, 84738, 1536867, 29687772, 603835479, 12831704772, 283320533673, 6473430313902, 152586247226958, 3701535783215857, 92238331155559794, 2357440730629390878, 61720161749858023305
Offset: 1

Views

Author

Paul D. Hanna, Jan 21 2009

Keywords

Examples

			G.f.: A(x) = x + 3*x^3 + 27*x^5 + 339*x^7 + 5067*x^9 +...
A(x)^3 = x^3 + 9*x^5 + 108*x^7 + 1530*x^9 + 24219*x^11 +...
A(x) = G(G(G(x))) where
G(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 +...
Let F(x) = g.f. of A153852 and H(x) = g.f. of A153854, then
A(x) = F(x) + x^2*H(x)^3 where
F(x) = x + 2*x^3 + 15*x^5 + 165*x^7 + 2213*x^9 +...
H(x) = x + 4*x^3 + 42*x^5 + 594*x^7 + 9827*x^9 +...
H(x)^3 = x^3 + 12*x^5 + 174*x^7 + 2854*x^9 + 51045*x^11 +...

Crossrefs

Cf. A153851, A153852, A153854, A153850.

Programs

  • PARI
    {a(n)=local(G=x+O(x^(2*n+1))); for(i=0, n, G=serreverse(x-G^3)); polcoeff(subst(G,x,subst(G,x,G)), 2*n-1)}

Formula

G.f.: A(x) = Sum_{n>=0} a(2n+1)*x^(2n+1) = G(G(G(x))) where G(x) is the g.f. of A153851.
G.f.: A(x) = F(x) + x^2*H(x)^3 where F(x) is the g.f. of A153852 and H(x) is the g.f. of A153854.

A153854 Nonzero coefficients of g.f.: A(x) = G(G(G(G(x)))) where G(x) = x + G(G(x))^3 is the g.f. of A153851.

Original entry on oeis.org

1, 4, 42, 594, 9827, 179928, 3545637, 73988631, 1618178067, 36832568283, 868184365137, 21113629246953, 528282055072773, 13569770211307323, 357215846155083585, 9623529095387448543, 265025641890780905892
Offset: 1

Views

Author

Paul D. Hanna, Jan 21 2009

Keywords

Examples

			G.f.: A(x) = x + 4*x^3 + 42*x^5 + 594*x^7 + 9827*x^9 +...
A(x)^3 = x^3 + 12*x^5 + 174*x^7 + 2854*x^9 + 51045*x^11 +...
A(x) = G(G(G(G(x)))) where
G(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 +...
A(x) = F(F(x)) where F(x) = G(G(x)) is the g.f. of A153852:
F(x) = x + 2*x^3 + 15*x^5 + 165*x^7 + 2213*x^9 + 33693*x^11 +...

Crossrefs

Cf. A153851, A153852, A153853, A153850.

Programs

  • PARI
    {a(n)=local(G=x+O(x^(2*n+1))); for(i=0, n, G=serreverse(x-G^3)); polcoeff(subst(subst(G,x,G),x,subst(G,x,G)), 2*n-1)}

Formula

G.f.: A(x) = Sum_{n>=0} a(2n+1)*x^(2n+1) = G(G(G(G(x)))) where G(x) is the g.f. of A153851.
G.f.: A(x) = F(F(x)) where F(x) is the g.f. of A153852.
Showing 1-4 of 4 results.

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