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 1-6 of 6 results.

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),", "))

A153852 Nonzero coefficients of g.f.: A(x) = G(G(x)) where G(x) = x + G(G(x))^3 is the g.f. of A153851.

Original entry on oeis.org

1, 2, 15, 165, 2213, 33693, 561867, 10053141, 190489374, 3788856192, 78613758564, 1693737431667, 37760673462507, 868775517322730, 20583609967109565, 501340716386677815, 12535093359045980151, 321360932709750239226
Offset: 1

Views

Author

Paul D. Hanna, Jan 21 2009

Keywords

Examples

			G.f.: A(x) = x + 2*x^3 + 15*x^5 + 165*x^7 + 2213*x^9 + ...
A(x)^3 = x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 + 145815*x^13 + ...
A(x) = G(G(x)) where
G(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 + ...
Also, A(x) = G(x) + G(G(G(x)))^3 where G(G(G(x))) begins
G(G(G(x))) = x + 3*x^3 + 27*x^5 + 339*x^7 + 5067*x^9 + 84738*x^11 + ... + A153853(n)*x^(2*n-1) + ...
G(G(G(x)))^3 = x^3 + 9*x^5 + 108*x^7 + 1530*x^9 + 24219*x^11 + ...

Crossrefs

Cf. A153851, A153853, 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,G), 2*n-1)}

Formula

G.f.: A(x) = Sum_{n>=0} a(2n+1)*x^(2n+1) = G(G(x)) where G(x) is the g.f. of A153851.
G.f.: A(x) = G(x) + G(G(G(x)))^3 where G(x) is the g.f. of A153851 and G(G(G(x))) is the g.f. of A153853.

Extensions

Formula corrected by Paul D. Hanna, Dec 07 2009

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.

A276364 G.f. A(x) satisfies: A(x - A(x)^3) = x + A(x)^3, where A(x) = Sum_{n>=1} a(n)*x^(2*n-1).

Original entry on oeis.org

1, 2, 18, 252, 4410, 88734, 1969668, 47104056, 1195658550, 31891944750, 887565934494, 25639389304560, 765765781572600, 23574635888791804, 746297727831434376, 24247096863466015152, 807243935471150901066, 27503153109167182217082, 957899411829034037383374, 34073454839478198669105444, 1236879534288183156526996062, 45788365378826408823663436974, 1727576456033196960394178300184
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^3 + 18*x^5 + 252*x^7 + 4410*x^9 + 88734*x^11 + 1969668*x^13 + 47104056*x^15 + 1195658550*x^17 + 31891944750*x^19 + 887565934494*x^21 + 25639389304560*x^23 + 765765781572600*x^25 +...
such that A(x - A(x)^3) = x + A(x)^3.
RELATED SERIES.
Note that Series_Reversion(x - A(x)^3) = x/2 + A(x)/2, which begins:
Series_Reversion(x - A(x)^2) = x + x^3 + 9*x^5 + 126*x^7 + 2205*x^9 + 44367*x^11 + 984834*x^13 + 23552028*x^15 + 597829275*x^17 + 15945972375*x^19 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 2*x^3 - 6*x^5 - 60*x^7 - 830*x^9 - 13950*x^11 - 267156*x^13 - 5629752*x^15 - 127807290*x^17 - 3082830030*x^19 - 78254901810*x^21 - 2076067799280*x^23 - 57266880966792*x^25 +...
then Series_Reversion(x + A(x)^3) = x/2 + R(x)/2.
Also, A(x) = x + 2 * A( x/2 + A(x)/2 )^3, where
A( x/2 + A(x)/2 ) = x + 3*x^3 + 33*x^5 + 528*x^7 + 10235*x^9 + 224001*x^11 + 5343738*x^13 + 136167888*x^15 + 3659113701*x^17 + 102800460825*x^19 + 3001057504233*x^21 + 90627712970220*x^23 + 2821487673544920*x^25 +...
and
A( x/2 + A(x)/2 )^3 = x^3 + 9*x^5 + 126*x^7 + 2205*x^9 + 44367*x^11 + 984834*x^13 + 23552028*x^15 + 597829275*x^17 + 15945972375*x^19 +...
which equals -x/2 + A(x)/2.

Links

Crossrefs

Cf. A153851, A275765.

Programs

  • Mathematica
    nmin = 1; nmax = 60; sol = {b[1] -> 1}; nsol = Length[sol];
Do[A[x_] = Sum[b[k] x^k, {k, 0, n}] /. sol;eq = CoefficientList[A[x - A[x]^3] - x - A[x]^3 + O[x]^(n+1), x][[nsol+1;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nsol + 1, nmax}];
a[n_] := b[2n-1];
a /@ Range[nmin, (nmax+1)/2 // Floor] /. sol (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, [0,0]); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - F^3) - F^3, #A) ); A[2*n-1]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 2 * A( x/2 + A(x)/2 )^3.
(2) A(x) = -x + 2 * Series_Reversion(x - A(x)^3).
(3) R(x) = -x + 2 * Series_Reversion(x + A(x)^3), where R(A(x)) = x.
(4) R( ( x/2 - R(x)/2 )^(1/3) ) = x/2 + R(x)/2, where R(A(x)) = x.

A276366 G.f. A(x) satisfies: A(x - A(x)^3) = x + A(x)^2.

Original entry on oeis.org

1, 1, 3, 12, 57, 300, 1697, 10126, 62991, 405247, 2680901, 18160444, 125562250, 883868590, 6321838520, 45869309028, 337167193262, 2508018933431, 18861358215299, 143293615189089, 1098997404472941, 8504070741463729, 66358269984208701, 521923129718567918, 4136089275165532156, 33013640650845937124
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 57*x^5 + 300*x^6 + 1697*x^7 + 10126*x^8 + 62991*x^9 + 405247*x^10 + 2680901*x^11 + 18160444*x^12 +...
such that A(x - A(x)^3) = x + A(x)^2.
RELATED SERIES.
A(x - A(x)^3) = x + x^2 + 2*x^3 + 7*x^4 + 30*x^5 + 147*x^6 + 786*x^7 + 4480*x^8 + 26814*x^9 + 166865*x^10 + 1072160*x^11 + 7076724*x^12 +...
which equals x + A(x)^2.

Links

Crossrefs

Cf. A153851, A275765, A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^3) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies: A'(x - A(x)^3) = (1 + 2*A'(x)*A(x)) / (1 - 3*A'(x)*A(x)^2).
Showing 1-6 of 6 results.

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