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.

Previous Showing 11-18 of 18 results.

A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).

Original entry on oeis.org

1, 2, -1, 3, -10, 36, -137, 543, -2219, 9285, -39587, 171369, -751236, 3328218, -14878455, 67030785, -304036170, 1387247580, -6363044315, 29323149825, -135700543190, 630375241380, -2938391049395, 13739779184085, -64430797069375, 302934667061301, -1427763630578197
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2023

Keywords

Crossrefs

Cf. A073157, A007863, A364336, A364337, A364338, A364339, A366326, A366327, A366328.
Cf. A000108, A002212, A112478.

Programs

  • Maple
    a := proc(n) option remember; if n = 1 then 2 elif n = 2 then -1 else (-3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2))/n fi; end:
seq(a(n), n = 1..30); # Peter Bala, Sep 10 2024
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(n-2, n-k)/(2*k-1));

Formula

G.f.: A(x) = -2*x*(1+x) / (1+x-sqrt((1+x)*(1+5*x))).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n-2,n-k)/(2*k-1).
a(n) ~ -(-1)^n * 5^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 07 2023
From Peter Bala, Sep 10 2024: (Start)
a(n) = 1/(1 - n) * Sum_{k = 0..n} binomial(-n+k, k)*binomial(-n+k+1, n-k) for n not equal to 1. Cf. A007863.
a(n) = Sum_{k = 0..n-2} binomial(-n+k+1, k)*binomial(-n+k+1, n-k)/(-n+k+1) for n >= 2.
P-recursive: n*a(n) = - 3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2) with a(1) = 2 and a(2) = -1. (End)

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).

A348957 G.f. A(x) satisfies A(x) = (1 + x * A(-x)) / (1 - x * A(x)).

Original entry on oeis.org

1, 2, 2, 10, 18, 98, 210, 1194, 2786, 16258, 39906, 236938, 601458, 3615330, 9399858, 57024426, 150947010, 922283522, 2475603138, 15212318730, 41290579410, 254909413218, 698230131858, 4327273358250, 11943274468770, 74260741616514, 206279837823650, 1286199407132554
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2021

Keywords

Crossrefs

Cf. A006318, A086246.
Cf. A349310, A361638, A364407, A366266, A366364.
Cf. A112478, A364394, A364395, A366363.

Programs

  • Mathematica
    nmax = 27; A[] = 0; Do[A[x] = (1 + x A[-x])/(1 - x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -(-1)^n a[n - 1] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 27}]
CoefficientList[y/.AsymptoticSolve[y-y^2+x(1+y^3)==0,y->1,{x,0,27}][[1]],x] (* Alexander Burstein, Nov 26 2021 *)

Formula

a(0) = 1; a(n) = -(-1)^n * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1).
a(n) ~ c * 3^(3*n/4) * (1 + sqrt(3))^n / (sqrt(2*Pi) * n^(3/2) * 2^(n/2)), where c = 3^(1/4) if n is even and c = (1 + sqrt(3))/sqrt(2) if n is odd. - Vaclav Kotesovec, Nov 14 2021
From Alexander Burstein, Nov 26 2021: (Start)
G.f.: A(-x) = 1/A(x).
G.f.: A(x) = 1 + x*(1+A(x)^3)/A(x). (End)
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n-3*k-2,n-1) for n > 0. - Seiichi Manyama, Apr 11 2024

A112477 Riordan array ((1-x+sqrt(1+6*x+x^2))/2, (sqrt(1+6*x+x^2)-x-1)/2).

Original entry on oeis.org

1, 1, 1, -2, -1, 1, 6, 2, -3, 1, -22, -6, 10, -5, 1, 90, 22, -38, 22, -7, 1, -394, -90, 158, -98, 38, -9, 1, 1806, 394, -698, 450, -194, 58, -11, 1, -8558, -1806, 3218, -2126, 978, -334, 82, -13, 1, 41586, 8558, -15310, 10286, -4942, 1838, -526, 110, -15, 1, -206098, -41586, 74614, -50746, 25150, -9922, 3142, -778, 142, -17, 1
Offset: 0

Views

Author

Paul Barry, Sep 07 2005

Keywords

Examples

			Triangle starts:
    1;
    1,  1;
   -2, -1,   1;
    6,  2,  -3,  1;
  -22, -6,  10, -5,  1;
   90, 22, -38, 22, -7, 1;
  ...

Crossrefs

Inverse of triangle A112475. Row sums are A112478.

Programs

  • Mathematica
    T[n_,k_]:=SeriesCoefficient[(1-x+Sqrt[1+6x+x^2])((Sqrt[1+6x+x^2]-x-1)/2)^k/2,{x,0,n}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, May 26 2024 *)

A371341 G.f. A(x) satisfies A(x) = 1 + x/A(x) * (1 + A(x)^5).

Original entry on oeis.org

1, 2, 6, 46, 330, 2778, 24094, 219318, 2048274, 19583410, 190497142, 1880184446, 18778814938, 189456108554, 1927852050830, 19763367194630, 203919590002210, 2116079501498722, 22069907395614182, 231222485352688590, 2432325883912444010
Offset: 0

Views

Author

Seiichi Manyama, Apr 12 2024

Keywords

Crossrefs

Cf. A112478, A348957, A364394, A364395, A366363, A371892.

Programs

  • Maple
    A371341 := proc(n)
    if n = 0 then
        1;
    else
        add(binomial(n,k)*binomial(2*n-5*k-2,n-1),k=0..n) ;
        (-1)^(n-1)*%/n ;
    end if;
end proc:
seq(A371341(n),n=0..60) ; # R. J. Mathar, Apr 22 2024
  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(2*n-5*k-2, n-1))/n);

Formula

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n-5*k-2,n-1) for n > 0.

A371892 G.f. A(x) satisfies A(x) = 1 + x/A(x) * (1 + A(x)^4).

Original entry on oeis.org

1, 2, 4, 24, 112, 688, 4032, 25856, 165888, 1103616, 7412480, 50699776, 350087168, 2444208128, 17198686208, 121945948160, 870026493952, 6242802761728, 45016506564608, 326071359897600, 2371312632397824, 17307835567636480, 126743329792327680
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2024

Keywords

Crossrefs

Cf. A112478, A348957, A364394, A364395, A366363.

Programs

  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(2*n-4*k-2, n-1))/n);

Formula

a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n-4*k-2,n-1) for n > 0.
Previous Showing 11-18 of 18 results.

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