A276310 -id:A276310 - cp's OEIS frontend

A276314 G.f. A(x) satisfies: x = A(x)-A(x)^2-3*A(x)^3.

1, 1, 5, 20, 104, 546, 3066, 17655, 104555, 630773, 3867617, 24020932, 150827740, 955808680, 6105327912, 39268000188, 254093573088, 1652984379150, 10804631902350, 70925539707330, 467373389649870, 3090558380977020, 20501504119375500, 136392970090612950

Offset: 1

Tom Richardson, Aug 29 2016

			G.f.: A(x) = x+x^2+5*x^3+20*x^4+104*x^5+546*x^6+3066*x^7+... Related Expansions:
A(x)^2=x^2+2*x^3+11*x^4+50*x^5+273*x^6+1500*x^7+8664*x^8+...
A(x)^3=x^3+3*x^4+18*x^5+91*x^6+522*x^7+2997*x^8+17831*x^9+...

Michael De Vlieger, Table of n, a(n) for n = 1..1181
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Cf. A250886, A276310, A276315, A276316.

Rest[CoefficientList[InverseSeries[Series[x - x^2 - 3*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)

{a(n)=polcoeff(serreverse(x - x^2 - 3*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

G.f.: Series_Reversion(x-x^2-3*x^3)
Conjecture: +169*n*(n+2)*(n-1)*a(n) +13*(n-1) *(13*n^2+26*n-220) *a(n-1) +(-7277*n^3+13423*n^2+43814*n-81700) *a(n-2) -27*(3*n-10) *(3*n-8) *(71*n+197)*a(n-3)=0. - R. J. Mathar, Sep 17 2016
a(n) ~ (29 + 20*sqrt(10))^(n - 1/2) / (2^(5/4) * 5^(1/4) * sqrt(Pi) * n^(3/2) * 13^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017

A276315 G.f. A(x) satisfies: x = A(x)-3A(x)^2-2A(x)^3.

Original entry on oeis.org

1, 3, 20, 165, 1524, 15078, 156264, 1674585, 18404980, 206325834, 2350049208, 27118926354, 316381296840, 3725407768140, 44217602683728, 528470024711841, 6354463541900148, 76818345766932450, 933089010748085400, 11382500895815005110, 139387948563917844120

Offset: 1

Tom Richardson, Aug 29 2016

			G.f.: A(x) = x+3*x^2+20*x^3+165*x^4+1524*x^5+15078*x^6+156264*x^7+...
Related Expansions:
A(x)^2 = x^2+6*x^3+49*x^4+450*x^5+4438*x^6+45900*x^7+491181*x^8+...
A(x)^3 = x^3+9*x^4+87*x^5+882*x^6+9282*x^7+100521*x^8+1113299*x^9+...

Michael De Vlieger, Table of n, a(n) for n = 1..897
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Cf. A250886, A276310, A276314, A276316.

Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 - 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)

{a(n)=polcoeff(serreverse(x - 3*x^2 - 2*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

G.f.: Series_Reversion(x-3*x^2-2*x^3).

a(n) ~ (6*(18 + 5*sqrt(15))/17)^(n - 1/2) / (2*15^(1/4)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 22 2017

A276316 G.f. A(x) satisfies: x = A(x)-4*A(x)^2+A(x)^3.

Original entry on oeis.org

1, 4, 31, 300, 3251, 37744, 459060, 5773548, 74474455, 979872036, 13099102575, 177414673488, 2429310288468, 33574008073120, 467717206216760, 6560977611629676, 92595131510426943, 1313820730347196300, 18730821529411507725, 268185082351558093260

Offset: 1

Tom Richardson, Aug 29 2016

			G.f.: A(x) = x+4*x^2+31*x^3+300*x^4+3251*x^5+37744*x^6+459060*x^7+...
Related Expansions:
A(x)^2 = x^2+8*x^3+78*x^4+848*x^5+9863*x^6+120096*x^7+1511634*x^8+...
A(x)^3 = x^3+12*x^4+141*x^5+1708*x^6+21324*x^7+272988*x^8+3566761*x^9+...

Robert Israel, Table of n, a(n) for n = 1..844
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Cf. A250886, A276310, A276314, A276315.

S:= series(RootOf(x-4*x^2+x^3-t,x),t,100):
seq(coeff(S,t,j),j=1..100); # Robert Israel, Sep 02 2016

Rest[CoefficientList[InverseSeries[Series[x - 4*x^2 + x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)

{a(n)=polcoeff(serreverse(x - 4*x^2 + x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

G.f.: Series_Reversion(x-4*x^2+x^3).
From Robert Israel, Sep 02 2016: (Start)
G.f. g(x) satisfies the differential equation
(12-184*t-27*t^2)*g''(t) - (92+27*t)*g'(t) + 3*g(t) = 4.
(-27*n^2+3)*a(n)+(-184*n^2-276*n-92)*a(n+1)+(12*n^2+36*n+24)*a(n+2) = 0
for n >= 1. (End)
a(n) ~ (46 + 13*sqrt(13))^(n - 1/2) / (13^(1/4) * sqrt(Pi) * n^(3/2) * 2^(n + 1/2) * 3^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017

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A276314 G.f. A(x) satisfies: x = A(x)-A(x)^2-3*A(x)^3.

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A276315 G.f. A(x) satisfies: x = A(x)-3A(x)^2-2A(x)^3.

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A276316 G.f. A(x) satisfies: x = A(x)-4*A(x)^2+A(x)^3.

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cp's OEIS Frontend

A276314 G.f. A(x) satisfies: x = A(x)-A(x)^2-3*A(x)^3.

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A276315 G.f. A(x) satisfies: x = A(x)-3*A(x)^2-2*A(x)^3.

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A276316 G.f. A(x) satisfies: x = A(x)-4*A(x)^2+A(x)^3.

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A276315 G.f. A(x) satisfies: x = A(x)-3A(x)^2-2A(x)^3.