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 1-5 of 5 results.

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

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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

A366454 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(3/2).

Original entry on oeis.org

1, 2, -3, 12, -58, 312, -1794, 10794, -67113, 427800, -2780677, 18360504, -122809416, 830379966, -5666465445, 38974338126, -269915089194, 1880576960904, -13172489198859, 92705253700620, -655219698720486, 4648722344211012, -33096948925057703
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366455, A366456.
Cf. A366400.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366400.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k/2-1,k) * binomial(n+3*k/2-2,n-k) / (5*k/2-1).

A366455 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(5/2).

Original entry on oeis.org

1, 2, -5, 30, -215, 1710, -14516, 128830, -1180920, 11093830, -106245975, 1033454774, -10181848705, 101394979530, -1018972470275, 10320779179380, -105250097458410, 1079767027094630, -11136159773691830, 115395278542757580, -1200814926210284360
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366454, A366456.
Cf. A366401.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366401.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(7*k/2-1,k) * binomial(n+5*k/2-2,n-k) / (7*k/2-1).

A366456 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(7/2).

Original entry on oeis.org

1, 2, -7, 56, -532, 5600, -62860, 737324, -8929726, 110811344, -1401640814, 18004922936, -234243536436, 3080152906096, -40870739065996, 546563064528906, -7358930622768977, 99672580921800656, -1357142384455626909, 18565841939010374736, -255054402946387767408
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366454, A366455.
Cf. A366402.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(9*k/2-1, k)*binomial(n+7*k/2-2, n-k)/(9*k/2-1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366402.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(9*k/2-1,k) * binomial(n+7*k/2-2,n-k) / (9*k/2-1).
Showing 1-5 of 5 results.

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