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.

A251690 G.f. A(x) satisfies the condition that G(A(x)) is a power series in x consisting entirely of positive integer coefficients such that G(A(x) - x^k) has negative coefficients for k>0, where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, -1, -2, -2, 0, -1, 0, -3, 0, -3, -3, 0, -3, -2, -3, -1, -2, 0, -1, -2, 0, 0, -2, 0, 0, 0, -2, 0, -3, 0, -2, 0, -1, 0, -3, -2, -1, -1, -3, -1, 0, -2, -2, -3, -1, -3, -1, -1, 0, 0, -1, -1, -3, -3, -1, 0, -1, 0, -2, 0, -3, -3, -3, -2, -1, -2, -1, -2, -2, -2, -3, -1, -3, -1, -3, -1, 0, -2, -2, -2, -1, -1, -2, -2, 0, -3, -3, -2, -3, -1, -3, -2, 0, 0, 0, -2, -2, -2, -2, -3, -3, 0, -2, 0, -3, -1, 0, -2, -3, -1, -3, 0, -1, 0, -2, -1, -1, -3, -3, -1, -3, -3, 0, -3, -2, -3, -2
Offset: 1

Views

Author

Paul D. Hanna, Dec 31 2014

Keywords

Comments

Compare to the similar series F(x) for the Catalan function C(x) = 1 + x*C(x)^2, where C(F(x)) consists entirely of positive integer coefficients such that C(F(x) - x^k) has negative coefficients for k>0; in which case F(x) = (x+x^2) - (x+x^2)^2, and C(F(x)) = 1/(1-x-x^2).
It seems that a(n) lies in [-3,0] for n>1; does this continue to hold?

Examples

			G.f.: A(x) = x - x^2 - 2*x^3 - 2*x^4 - x^6 - 3*x^8 - 3*x^10 - 3*x^11 - 3*x^13 - 2*x^14 - 3*x^15 - x^16 - 2*x^17 - x^19 - 2*x^20 - 2*x^23 - 2*x^27 - 3*x^29 +...
Given G(x) = 1 + x*G(x)^3, which begins
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 +...
then
G(A(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 36*x^6 + 78*x^7 + 169*x^8 + 370*x^9 + 813*x^10 + 1793*x^11 + 3971*x^12 +...+ A251691(n)*x^n +...
consists entirely of positive integer coefficients such that
G(A(x) - x^k) has negative coefficients for k>0, as illustrated by:
k=1: G(A(x) - x) = 1 - x^2 - 2*x^3 + x^4 + 12*x^5 + 11*x^6 - 48*x^7 +...
k=2: G(A(x) - x^2) = 1 + x + x^2 - 2*x^3 - 19*x^4 - 83*x^5 - 267*x^6 +...
k=3: G(A(x) - x^3) = 1 + x + 2*x^2 + 3*x^3 + 2*x^4 - 13*x^5 - 97*x^6 - 471*x^7 +...
k=4: G(A(x) - x^4) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 6*x^6 - 58*x^7 - 413*x^8 +...
k=5: G(A(x) - x^5) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 30*x^6 + 48*x^7 + 33*x^8 - 215*x^9 - 1632*x^10 +...
k=6: G(A(x) - x^6) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 35*x^6 + 72*x^7 + 139*x^8 + 234*x^9 + 228*x^10 - 655*x^11 - 6102*x^12 +...
etc.
The position of the first negative term in G(A(x) - x^n), n>=1, begins:
[2, 3, 5, 7, 9, 11, 13, 16, 18, 20, 23, 25, 28, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 59, 62, 64, 67, 69, 72, 74, 77, 79, 82, 84, 87, 89, 92, 94, 97, 99, 102, 104, 106, 109, 111, 114, 116, 119, ...];
does the ratio of the position of the first negative term in G(A(x)-x^n) divided by n approach some constant?

Links

Crossrefs

Cf. A251691, A001764.

Programs

  • PARI
    /* Prints initial N terms: */
N=100;
/* G(x) = 1 + x*G(x)^3 is the g.f. of A001764: */
{G=1+serreverse(x/(1+x +x*O(x^(3*N+10)))^3);}
/* Print terms as you build vector A, then print A at the end: */
{A=[1,-1];print1("1, -1, ");
for(l=1,N,A=concat(A,-4);
for(i=1,4,A[#A]=A[#A]+1;
V=Vec(subst(G,x,x*truncate(Ser(A)) +O(x^floor(3*#A+1)) ));
if((sign(V[3*#A])+1)/2==1, print1(A[#A],", "); break));); A}

A251571 G.f.: M(F(x)) is a power series in x consisting entirely of positive integer coefficients such that M(F(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + x*M(x) + x*M(x)^2 is the g.f. of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 19, 27, 39, 55, 79, 113, 160, 228, 322, 455, 641, 902, 1268, 1777, 2490, 3483, 4864, 6791, 9468, 13189, 18358, 25527, 35473, 49248, 68336, 94751, 131288, 181815, 251627, 348051, 481180, 664885, 918285, 1267663, 1749212, 2412635, 3326303, 4584236, 6315428, 8697260
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2015

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 13*x^7 +...
such that A(x) = M(F(x)),
where F(x) is the g.f. of A251570:
F(x) = x - x^3 - x^4 + x^5 - x^7 - x^8 + x^10 - x^11 - x^13 - x^14 - x^16 - x^17 - x^18 - x^20 - x^22 - x^26 - x^27 - x^28 - x^29 - x^32 - x^33 - x^35 - x^36 - x^39 - x^41 - x^43 - x^44 - x^45 - x^46 - x^47 - x^48 - x^50 +...
and M(x) is the g.f. of the Motzkin numbers:
M(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 + 323*x^8 + 835*x^9 + 2188*x^10 + 5798*x^11 + 15511*x^12 +...

Crossrefs

Cf. A251570, A251691.

Programs

  • PARI
    /* Prints initial N+2 terms: */
N=100;
/* M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of Motzkin numbers: */
{M=1/x*serreverse(x/(1+x+x^2 +x*O(x^(2*N+10)))); M +O(x^21) }
/* Print terms as you build vector A, then print a(n) at the end: */
{A=[1, 0]; print1("1, 0, ");
for(l=1, N, A=concat(A, -3);
for(i=1, 4, A[#A]=A[#A]+1;
V=Vec(subst(M, x, x*truncate(Ser(A)) +O(x^floor(2*#A+1)) ));
if((sign(V[2*#A])+1)/2==1, print1(A[#A], ", "); break)); );
Vec(subst(M,x,x*Ser(A)))}

A257084 G.f. A(x) satisfies A(F(x)) = x, where F(x) is the g.f. of A251690.

Original entry on oeis.org

1, 1, 4, 17, 80, 407, 2160, 11859, 66754, 383210, 2234921, 13204685, 78870454, 475453371, 2888991879, 17675743626, 108801199823, 673302178725, 4186513098755, 26142455226568, 163873586066647, 1030820865387599, 6504789754356175, 41166205256238155, 261217480924768212, 1661598566523216015
Offset: 1

Views

Author

Paul D. Hanna, Apr 15 2015

Keywords

Comments

G.f. F(x) of A251690 satisfies the condition that G(F(x)) is a power series in x consisting entirely of positive integer coefficients such that G(F(x) - x^k) has negative coefficients for k>0, where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 17*x^4 + 80*x^5 + 407*x^6 + 2160*x^7 +...
such that the series reversion of A(x) yields the g.f. F(x) of A251690:
F(x) = x - x^2 - 2*x^3 - 2*x^4 - x^6 - 3*x^8 - 3*x^10 - 3*x^11 - 3*x^13 - 2*x^14 - 3*x^15 - x^16 - 2*x^17 - x^19 - 2*x^20 - 2*x^23 - 2*x^27 - 3*x^29 - 2*x^31 - x^33 - 3*x^35 - 2*x^36 - x^37 - x^38 - 3*x^39 - x^40 - 2*x^42 - 2*x^43 - 3*x^44 - x^45 - 3*x^46 - x^47 - x^48 - x^51 -...
in which all coefficients after the first are in the interval [-3,0].
RELATED SERIES.
Given G(x) = 1 + x*G(x)^3, which begins
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 +...
then
G(F(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 36*x^6 + 78*x^7 + 169*x^8 + 370*x^9 + 813*x^10 + 1793*x^11 + 3971*x^12 +...+ A251691(n)*x^n +...
consists entirely of positive integer coefficients such that G(F(x) - x^k) has negative coefficients for k>0.
Also, a related series is defined by the limits:
1/F'(x) = Limit ( A(F(x) + x^n) - x ) / x^n, and
1/F'(x) = Limit ( x - A(F(x) - x^n) ) / x^n, where
1/F'(x) = 1 + 2*x + 10*x^2 + 40*x^3 + 156*x^4 + 638*x^5 + 2544*x^6 + 10248*x^7 + 41152*x^8 + 165350*x^9 + 664477*x^10 + 2669644*x^11 + 10727319*x^12 + 43102392*x^13 + 173188681*x^14 + 695884096*x^15 + 2796104790*x^16 +...

Crossrefs

Cf. A251690, A251691, A001764.

Programs

  • PARI
    /* Prints initial N terms: */
N=50;
/* G(x) = 1 + x*G(x)^3 is the g.f. of A001764: */
{G=1+serreverse(x/(1+x +x*O(x^(3*N+10)))^3); }
/* Build the series reversion, then print coefficients at end: */
{A=[1, -1]; for(l=1, N, A=concat(A, -4);
for(i=1, 4, A[#A]=A[#A]+1;
V=Vec(subst(G, x, x*truncate(Ser(A)) +O(x^floor(3*#A+1)) ));
if((sign(V[3*#A])+1)/2==1, print1(".");break)););
Vec(serreverse(x*Ser(A)))}
Showing 1-3 of 3 results.

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