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.

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Nov 08 2015

Keywords

Comments

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).

Examples

			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 +...

Links

Crossrefs

Cf. A011791, A295537, A295538, A366452.
Cf. A370472, A370473, A370476.

Programs

  • PARI
    {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),", "))

Formula

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)

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)

A366453 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^(7/2).

Original entry on oeis.org

1, 2, 7, 42, 287, 2142, 16898, 138600, 1170037, 10098774, 88712736, 790540296, 7128879940, 64933227996, 596523624144, 5520761026854, 51424824505054, 481741853731110, 4535711525840271, 42897532229559714, 407358615638833341, 3882484733036731500
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366454, A366455, A366456.
Cf. A295537, A366405.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*k/2+1, n-k)*binomial(7*k/2, k)/(5*k/2+1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366405.
a(n) = Sum_{k=0..n} binomial(5*k/2+1,n-k) * binomial(7*k/2,k) / (5*k/2+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A295537. - 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).

A011792 Number of directed animals on a certain lattice.

Original entry on oeis.org

1, 3, 18, 121, 896, 7028, 57406, 483080
Offset: 0

Views

Author

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

Keywords

Crossrefs

Cf. A295537.

Formula

a(n) = A295537(n+1), at least for 0 <= n <= 7, (the known terms). - Omar E. Pol, Dec 05 2017
Showing 1-7 of 7 results.

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