A251690 -id:A251690 - cp's OEIS frontend

A251691 G.f.: 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 and F(x) is g.f. of A251690.

1, 1, 2, 4, 8, 17, 36, 78, 169, 370, 813, 1793, 3971, 8817, 19631, 43804, 97938, 219357, 492072, 1105398, 2486320, 5598805, 12620832, 28477139, 64311189, 145354456, 328772330, 744155150, 1685434388, 3819629781, 8661130303, 19649713303, 44601771038, 101285994072, 230110466746

Offset: 0

Paul D. Hanna, Dec 31 2014

nonn

Paul D. Hanna, Table of n, a(n) for n = 0..120

Cf. A251690, A001764.

G.f.: 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 +...
such that A(x) = G(F(x)), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 +...
and F(x) is the g.f. 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 +...

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

Paul D. Hanna, Apr 15 2015

nonn

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.

			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 +...

Cf. A251690, A251691, A001764.

/* 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)))}

A251570 G.f. A(x) satisfies the condition that M(A(x)) is a power series in x consisting entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + xM(x) + xM(x)^2 is the g.f. of the Motzkin numbers A001006.

Original entry on oeis.org

1, 0, -1, -1, 1, 0, -1, -1, 0, 1, -1, 0, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, 0, 0, 0, -1, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, -1, 0, -1, -1, -1, 0, -1, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -2, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Paul D. Hanna, Jan 06 2015

sign

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).

			G.f.: A(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 +...
Given the g.f. M(x) 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 +...
then
M(A(x)) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 13*x^7 + 19*x^8 + 27*x^9 + 39*x^10 + 55*x^11 + 79*x^12 + 113*x^13 + 160*x^14 +...+ A251571(n)*x^n +...
consists entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0.
Note that a(n) = -2 seems somewhat sparse and occurs at positions:
[58, 123, 181, 187, 203, 213, 230, 236, 245, 253, ...].

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A251571, A251690, A001006.

/* Prints initial N 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 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)); ); A}

cp's OEIS Frontend

A251691 G.f.: 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 and F(x) is g.f. of A251690.

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

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A251570 G.f. A(x) satisfies the condition that M(A(x)) is a power series in x consisting entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + xM(x) + xM(x)^2 is the g.f. of the Motzkin numbers A001006.

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cp's OEIS Frontend

A251691 G.f.: 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 and F(x) is g.f. of A251690.

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

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PARI

A251570 G.f. A(x) satisfies the condition that M(A(x)) is a power series in x consisting entirely of positive integer coefficients such that M(A(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.

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PARI

A251570 G.f. A(x) satisfies the condition that M(A(x)) is a power series in x consisting entirely of positive integer coefficients such that M(A(x) - x^k) has negative coefficients for k>0, where M(x) = 1 + xM(x) + xM(x)^2 is the g.f. of the Motzkin numbers A001006.