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 1-7 of 7 results.

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

Views

Author

Paul D. Hanna, Nov 27 2017

Keywords

Comments

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).

Examples

			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).

Links

Crossrefs

Cf. A011792, A259757, A295538.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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), ", "))
  • PARI
    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

Formula

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)

A295538 G.f. A(x) satisfies A(x)^2 = 1 + x + x*A(x)^9.

Original entry on oeis.org

1, 1, 4, 32, 290, 2894, 30624, 337602, 3835395, 44588657, 527903344, 6343105788, 77153875396, 948150877136, 11754481411170, 146829606548967, 1846232392749705, 23349436820785896, 296822925777158448, 3790612373731979898, 48608130217245939310, 625636961746371994680, 8079794260209350950338, 104667769434155291997329, 1359712949654853908780859, 17709395639599543591065564
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2017

Keywords

Comments

Note that the function G(x) = 1 + x*G(x)^4 (g.f. of A002293) also satisfies the condition: G(x) = 1/G(-x*G(x)^7).

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 290*x^4 + 2894*x^5 + 30624*x^6 + 337602*x^7 + 3835395*x^8 + 44588657*x^9 + 527903344*x^10 + 6343105788*x^11 + 77153875396*x^12 + 948150877136*x^13 + 11754481411170*x^14 + 146829606548967*x^15 +...
such that A(x)^2 = 1+x + x*A(x)^9.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 9*x^2 + 72*x^3 + 660*x^4 + 6624*x^5 + 70380*x^6 + 778164*x^7 + 8860302*x^8 + 103187376*x^9 + 1223410846*x^10 +...
A(x)^9 = 1 + 9*x + 72*x^2 + 660*x^3 + 6624*x^4 + 70380*x^5 + 778164*x^6 + 8860302*x^7 + 103187376*x^8 + 1223410846*x^9 + 14717253672*x^10 +...
A(-x*A(x)^7) = 1 - x - 3*x^2 - 25*x^3 - 221*x^4 - 2187*x^5 - 22989*x^6 - 252237*x^7 - 2855304*x^8 - 33101152*x^9 - 391010608*x^10 +...
which equals 1/A(x).

Links

Crossrefs

Cf. A259757, A295537, A366453.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A = sqrt(1+x + x*A^9 +x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = 1 + Series_Reversion( x/(1 + 4*x + 16*x^2 + 34*x^3 + 46*x^4 + 40*x^5 + 22*x^6 + 7*x^7 + x^8) ).
(2) F(A(x)) = x such that F(x) = -(1 - x^2)/(1 + x^9).
(3) A(x) = 1 / A(-x*A(x)^7).
a(n) ~ sqrt((1 + s^9)/(7*Pi)) / (2*n^(3/2)*r^(n - 1/2)), where r = 0.07223758934231429961770532152600550503126361567079... and s = 1.174134228398636389214738979941451774138268651734... are real roots of the system of equations 1 + r + r*s^9 = s^2, 9*r*s^7 = 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(x)^7 + A(x)^8).
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(9*k/2+1/2,n)/(9*k+1). (End)

A366452 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^(5/2).

Original entry on oeis.org

1, 2, 5, 20, 90, 440, 2266, 12110, 66525, 373320, 2130865, 12332512, 72202860, 426861830, 2544727475, 15280236800, 92333523153, 561054410200, 3426075429740, 21013974400920, 129403499560500, 799733464576880, 4958649842375975, 30837325310579350
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366453, A366454, A366455, A366456.
Cf. A259757, A366404.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*k/2+1, n-k)*binomial(5*k/2, k)/(3*k/2+1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366404.
a(n) = Sum_{k=0..n} binomial(3*k/2+1,n-k) * binomial(5*k/2,k) / (3*k/2+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A259757. - Seiichi Manyama, Apr 04 2024

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

Views

Author

Paul D. Hanna, Nov 27 2017

Keywords

Comments

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).

Examples

			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 +...

Links

Crossrefs

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

Programs

  • PARI
    {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), ", "))
  • PARI
    {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), ", "))

Formula

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

Views

Author

Paul D. Hanna, Nov 27 2017

Keywords

Examples

			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).

Links

Crossrefs

Cf. A295808. A259757, A295537, A295538.

Programs

  • PARI
    {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), ", "))
  • PARI
    {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), ", "))

Formula

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

Views

Author

Paul D. Hanna, Nov 27 2017

Keywords

Comments

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).

Examples

			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 +...

Links

Crossrefs

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

Programs

  • PARI
    {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), ", "))
  • PARI
    {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), ", "))

Formula

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).

A011791 Number of directed animals on a certain lattice.

Original entry on oeis.org

1, 2, 8, 35, 169, 862, 4575, 24999, 139700, 794684, 4586377, 26788423, 158054285, 940603900
Offset: 0

Views

Author

Alain Denise (Alain.Denise(AT)lri.fr)

Keywords

Links

Crossrefs

Cf. A259757.
Showing 1-7 of 7 results.

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