A295538 -id:A295538 - cp's OEIS frontend

A259757 G.f. A(x) satisfies A(x)^2 = 1 +x + x*A(x)^5.

1, 1, 2, 8, 35, 169, 862, 4575, 24999, 139700, 794684, 4586377, 26788423, 158054285, 940603900, 5639481930, 34032324940, 206550445064, 1259975808104, 7720835953740, 47504293931640, 293357473042545, 1817649401577760, 11296505623845080, 70402438290940450, 439888817329463279, 2755010697928837222, 17292270772076728414

Offset: 0

Paul D. Hanna, Nov 08 2015

nonn

Terms appear to equal A011791, apart from offset and an initial 1.

Note that the function G(x) = 1 + x*G(x)^2 (g.f. of A000108) also satisfies this condition: G(x) = 1/G(-x*G(x)^3).

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 35*x^4 + 169*x^5 + 862*x^6 + 4575*x^7 + 24999*x^8 + 139700*x^9 + 794684*x^10 +...
where A(x)^2 = 1+x + x*A(x)^5 and
A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 90*x^4 + 440*x^5 + 2266*x^6 + 12110*x^7 + 66525*x^8 + 373320*x^9 + 2130865*x^10 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 90*x^3 + 440*x^4 + 2266*x^5 + 12110*x^6 + 66525*x^7 + 373320*x^8 + 2130865*x^9 + 12332512*x^10 +...
OTHER RELATIONS.
Let B(x) be defined by B(x*A(x)) = x, then
B(x) =  x - x^2 - 3*x^4 - 3*x^5 - 22*x^6 - 50*x^7 - 240*x^8 - 763*x^9 - 3234*x^10 - 11880*x^11 - 48831*x^12 +...
Let C(x) be defined by C(x*A(x)^2) = A(x), then
C(x) = 1 + x + 3*x^3 - 3*x^4 + 22*x^5 - 50*x^6 + 240*x^7 - 763*x^8 + 3234*x^9 - 11880*x^10 + 48831*x^11 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..530

Cf. A011791, A295537, A295538, A366452.

Cf. A370472, A370473, A370476.

{a(n) = my(A=1+x); for(i=1,n, A = sqrt(1+x + x*A^5 +x*O(x^n)) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies [from Paul D. Hanna, Nov 27 2017]:
(1) 1 + Series_Reversion( x/(1 + 2*x + 4*x^2 + 3*x^3 + x^4) ).
(2) F(A(x)) = x such that F(x) = -(1 - x^2)/(1 + x^5).
(3) A(x) = 1 / A(-x*A(x)^3).
Recurrence: 3*(n-2)*(n-1)*n*(3*n - 1)*(3*n + 1)*a(n) = 6*(n-2)*(n-1)*(2*n - 1)*(3*n - 2)*(3*n - 1)*a(n-1) + 10*(n-2)*(41*n^4 - 164*n^3 + 200*n^2 - 72*n + 3)*a(n-2) + 100*(n-3)*n*(2*n - 3)*(2*n^2 - 6*n + 3)*a(n-3) + 125*(n-4)*(n-3)*(n-1)^2*n*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ 3^(n - 5/2) * 5^n * sqrt((15 + 4*10^(1/3) + 2*10^(2/3))/Pi) / (2*n^(3/2) * (10^(2/3) + 4*10^(1/3) - 11)^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017
D-finite with recurrence 9*n*(3*n-1)*(3*n+1)*a(n) -6*(3*n-2) *(48*n^2-115*n+83)*a(n-1) +15*(n-1) *(17*n^2-169*n+254)*a(n-2) +50 *(n-3)*(194*n^2-971*n+1200) *a(n-3) +125*(n-4) *(143*n^2-856*n+1265) *a(n-4) +2500*(n-5) *(5*n^2-35*n+59)*a(n-5) +3125*(n-5)*(n-6)*(n-3)*a(n-6)=0. - R. J. Mathar, Nov 16 2023
From Seiichi Manyama, Apr 04 2024: (Start)
G.f. A(x) satisfies A(x) = 1 + x * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4).
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k/2+1/2,n)/(5*k+1). (End)

A295537 G.f. A(x) satisfies A(x)^2 = 1 + x + x*A(x)^7.

Original entry on oeis.org

1, 1, 3, 18, 121, 896, 7028, 57406, 483080, 4159169, 36462855, 324391132, 2921210383, 26576350332, 243901358678, 2255283941595, 20991223674553, 196508265126327, 1849038158249933, 17478100523106657, 165891345107764059, 1580380321767062796, 15106335141526197636, 144839560162346664092, 1392621873057558622860, 13424503737125805253734

Offset: 0

Paul D. Hanna, Nov 27 2017

nonn

Terms appear to equal A011792, apart from offset and an initial 1.

Note that the function G(x) = 1 + x*G(x)^3 (g.f. of A001764) also satisfies the condition: G(x) = 1/G(-x*G(x)^5).

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 121*x^4 + 896*x^5 + 7028*x^6 + 57406*x^7 + 483080*x^8 + 4159169*x^9 + 36462855*x^10 + 324391132*x^11 + 2921210383*x^12 + 26576350332*x^13 + 243901358678*x^14 + 2255283941595*x^15 + ...
such that A(x)^2 = 1+x + x*A(x)^7.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 7*x^2 + 42*x^3 + 287*x^4 + 2142*x^5 + 16898*x^6 + 138600*x^7 + 1170037*x^8 + 10098774*x^9 + 88712736*x^10 + ...
A(x)^7 = 1 + 7*x + 42*x^2 + 287*x^3 + 2142*x^4 + 16898*x^5 + 138600*x^6 + 1170037*x^7 + 10098774*x^8 + 88712736*x^9 + ...
A(-x*A(x)^5) = 1 - x - 2*x^2 - 13*x^3 - 84*x^4 - 616*x^5 - 4788*x^6 - 38865*x^7 - 325489*x^8 - 2791845*x^9 - 24401730*x^10 + ...
which equals 1/A(x).

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A011792, A259757, A295538.

a:= n-> coeff(series(RootOf(x*_Z^6-x*_Z^5+x*_Z^4-x*_Z^3
        +x*_Z^2-(1+x)*_Z+1+x), x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 06 2017

m = 26; A[_] = 0;
Do[A[x_] = Sqrt[1 + x + x A[x]^7] + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 02 2019 *)

{a(n) = my(A=1+x); for(i=1, n, A = sqrt(1+x + x*A^7 +x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A295537(N=20)=Vec(serreverse('x/Ser(Polrev([1,3,9,13,11,5,1]),,N))+1) \\ Yields a vector with N terms. To compute only a(n) use polcoeff(...,n) instead of Vec(), and N = n+1. - M. F. Hasler, Mar 16 2018

G.f. A(x) satisfies:
(1) 1 + Series_Reversion( x/(1 + 3*x + 9*x^2 + 13*x^3 + 11*x^4 + 5*x^5 + x^6) ).
(2) F(A(x)) = x such that F(x) = -(1 - x^2)/(1 + x^7).
(3) A(x) = 1 / A(-x*A(x)^5).
a(n) ~ sqrt((1 + s^7)/(5*Pi)) / (2*n^(3/2)*r^(n - 1/2)), where r = 0.09760952485001949704447959455483068408354114893087... and s = 1.239618221385127629837391742046923301758368685253... are real roots of the system of equations 1 + r + r*s^7 = s^2, 7*r*s^5 = 2. - Vaclav Kotesovec, Nov 28 2017
From Seiichi Manyama, Apr 04 2024: (Start)
G.f. A(x) satisfies A(x) = 1 + x * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4 - A(x)^5 + A(x)^6).
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7*k/2+1/2,n)/(7*k+1). (End)

A295808 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^4 + A(x)^8).

Original entry on oeis.org

1, 1, 3, 17, 110, 783, 5908, 46433, 376029, 3115941, 26293410, 225166050, 1951877304, 17094430060, 151026790086, 1344405191931, 12046557221374, 108569555036541, 983512740523989, 8950335155129326, 81786796816686222, 750133309106091800, 6903286130025559800, 63724450582843480092, 589897018021520290940, 5474784437150040712036, 50932035767512193052753, 474865449870890392910894

Offset: 0

Paul D. Hanna, Nov 27 2017

nonn

Compare to: G(x)^3 = 1+x + x*(G(x)^2 + G(x)^4) holds when G(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).

			G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 110*x^4 + 783*x^5 + 5908*x^6 + 46433*x^7 + 376029*x^8 + 3115941*x^9 + 26293410*x^10 + 225166050*x^11 + 1951877304*x^12 +...
such that A(x)^3 = 1+x + x*(A(x)^4 + A(x)^8).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 12*x^2 + 70*x^3 + 468*x^4 + 3393*x^5 + 25932*x^6 + 205716*x^7 + 1677804*x^8 + 13980710*x^9 + 118505772*x^10 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 108*x^3 + 735*x^4 + 5388*x^5 + 41496*x^6 + 331036*x^7 + 2711511*x^8 + 22670964*x^9 + 192695140*x^10 +...
A(x)^8 = 1 + 8*x + 52*x^2 + 360*x^3 + 2658*x^4 + 20544*x^5 + 164220*x^6 + 1346768*x^7 + 11269199*x^8 + 95834808*x^9 + 825905828*x^10 +...

Paul D. Hanna, Table of n, a(n) for n = 0..520

Cf. A295809, A295810. A259757, A295537, A295538.

{a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^4 + x*A^8 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 3*x + 8*x^2 + 11*x^3 + 10*x^4 + 5*x^5 + x^6 +x*O(x^n)) ); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) =  1 + Series_Reversion( x/(1 + 3*x + 8*x^2 + 11*x^3 + 10*x^4 + 5*x^5 + x^6) ).
(2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^3 - x^5 + x^6).
(3) A(x) satisfies: A(x) = 1/A(-x*A(x)^5).

A295810 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^5 + A(x)^7).

Original entry on oeis.org

1, 1, 3, 16, 97, 645, 4539, 33242, 250715, 1934131, 15190377, 121050779, 976334857, 7954909796, 65378035310, 541346845867, 4511820592102, 37819912868231, 318639423484669, 2696819879180630, 22918021992024063, 195480539889732302, 1672951468057552136, 14361120982757852887, 123625070638172667688, 1066939084577136857174

Offset: 0

Paul D. Hanna, Nov 27 2017

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 97*x^4 + 645*x^5 + 4539*x^6 + 33242*x^7 + 250715*x^8 + 1934131*x^9 + 15190377*x^10 + 121050779*x^11 + 976334857*x^12 +...
such that A(x)^3 = 1+x + x*(A(x)^5 + A(x)^7).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 12*x^2 + 67*x^3 + 423*x^4 + 2880*x^5 + 20607*x^6 + 152763*x^7 + 1162908*x^8 + 9037195*x^9 + 71398917*x^10 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 150*x^3 + 990*x^4 + 6936*x^5 + 50640*x^6 + 381070*x^7 + 2934665*x^8 + 23016905*x^9 + 183216323*x^10 +...
A(x)^7 = 1 + 7*x + 42*x^2 + 273*x^3 + 1890*x^4 + 13671*x^5 + 102123*x^6 + 781838*x^7 + 6102530*x^8 + 48382012*x^9 + 388548244*x^10 +...
Series_Reversion(A(x) - 1) = x - 3*x^2 + 2*x^3 + 8*x^4 - 21*x^5 + 4*x^6 + 74*x^7 - 137*x^8 - 59*x^9 + 623*x^10 - 797*x^11 - 1083*x^12 + 4840*x^13 - 3793*x^14 - 12355*x^15 +...
which equals x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5).

Paul D. Hanna, Table of n, a(n) for n = 0..520

Cf. A295808. A259757, A295537, A295538.

{a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^5 + x*A^7 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5 +x*O(x^n)) ); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) = 1 + Series_Reversion( x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5) ).
(2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^3 - x^4 + x^5).

A295809 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^2 + A(x)^7).

Original entry on oeis.org

1, 1, 2, 9, 44, 238, 1363, 8129, 49947, 313982, 2009804, 13054923, 85835763, 570162938, 3820449453, 25792692527, 175277931388, 1198017908942, 8230391249548, 56801549666858, 393622890736512, 2737841490099777, 19107124975145342, 133755761621788177, 938960164858527807, 6608463199584560132, 46621379983243723382, 329627663780846842009

Offset: 0

Paul D. Hanna, Nov 27 2017

nonn

Compare to: G(x)^3 = 1+x + x*(G(x)^2 + G(x)^4) holds when G(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 44*x^4 + 238*x^5 + 1363*x^6 + 8129*x^7 + 49947*x^8 + 313982*x^9 + 2009804*x^10 + 13054923*x^11 + 85835763*x^12 +...
such that A(x)^3 = 1+x + x*(A(x)^2 + A(x)^7).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 9*x^2 + 40*x^3 + 204*x^4 + 1125*x^5 + 6536*x^6 + 39390*x^7 + 243966*x^8 + 1543350*x^9 + 9929589*x^10 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 110*x^4 + 600*x^5 + 3459*x^6 + 20728*x^7 + 127824*x^8 + 805852*x^9 + 5170270*x^10 +...
A(x)^7 = 1 + 7*x + 35*x^2 + 182*x^3 + 1015*x^4 + 5936*x^5 + 35931*x^6 + 223238*x^7 + 1415526*x^8 + 9123737*x^9 + 59601227*x^10 +...

Paul D. Hanna, Table of n, a(n) for n = 0..520

Cf. A295808, A295810. A259757, A295537, A295538.

{a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^2 + x*A^7 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 2*x + 5*x^2 + 6*x^3 + 4*x^4 + x^5 +x*O(x^n)) ); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) = 1 + Series_Reversion( x/(1 + 2*x + 5*x^2 + 6*x^3 + 4*x^4 + x^5) ).
(2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^2 - x^4 + x^5).

cp's OEIS Frontend

A259757 G.f. A(x) satisfies A(x)^2 = 1 +x + x*A(x)^5.

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A295537 G.f. A(x) satisfies A(x)^2 = 1 + x + x*A(x)^7.

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A295808 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^4 + A(x)^8).

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A295810 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^5 + A(x)^7).

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A295809 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^2 + A(x)^7).

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