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 1-10 of 10 results.

A366364 G.f. satisfies A(x) = (1 + x/A(x)^2)/(1 - x).

Original entry on oeis.org

1, 2, -2, 14, -70, 426, -2714, 18118, -124814, 881042, -6339058, 46318334, -342769750, 2563781690, -19350683018, 147197511222, -1127334112542, 8685458120226, -67270210217186, 523472089991662, -4090668558473318, 32088204418069450, -252576222775705466
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366365, A366366.
Cf. A366357.

Programs

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

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(3*k-1,k) * binomial(2*k-1,n-k)/(3*k-1).

A366403 G.f. A(x) satisfies A(x) = (1 + x / sqrt(A(x))) / (1 - x).

Original entry on oeis.org

1, 2, 1, 2, 0, 4, -5, 16, -35, 92, -231, 604, -1584, 4214, -11297, 30538, -83096, 227476, -625991, 1730788, -4805594, 13393690, -37458329, 105089230, -295673993, 834086422, -2358641375, 6684761126, -18985057350, 54022715452, -154000562757, 439742222072
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A366363, A366400, A366401, A366402, A366404, A366405, A366406, A366407.

Programs

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

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(3*k/2-1,k) * binomial(k/2-1,n-k) / (3*k/2-1).

A366356 G.f. satisfies A(x) = 1/(1 - x) + x/A(x).

Original entry on oeis.org

1, 2, -1, 6, -17, 71, -292, 1284, -5807, 26961, -127627, 613815, -2990680, 14730714, -73229290, 366936232, -1851352819, 9397497759, -47957377933, 245903408245, -1266266092111, 6545667052321, -33954266444497, 176689391245147, -922112642288148, 4825154135801698
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Links

Crossrefs

Partial sums give A363816.
Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366357, A366358, A366359.
Cf. A112478, A366363.

Programs

  • Mathematica
    A366356[n_]:=(-1)^(n-1)Sum[Binomial[2k-1,k]Binomial[2k-1,n-k]/(2k-1),{k,0,n}];
Array[A366356,30,0] (* Paolo Xausa, Oct 20 2023 *)
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(2*k-1, n-k)/(2*k-1));

Formula

G.f.: A(x) = -2*x*(1-x) / (1-sqrt(1+4*x*(1-x)^2)).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(2*k-1,n-k)/(2*k-1).

A366365 G.f. satisfies A(x) = (1 + x/A(x)^3)/(1 - x).

Original entry on oeis.org

1, 2, -4, 32, -240, 2064, -18816, 179264, -1762816, 17758976, -182342400, 1901196288, -20075427840, 214246524928, -2307200135168, 25039992254464, -273603550461952, 3007387399258112, -33230774508716032, 368915340555517952, -4112806343370539008
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366364, A366366.
Cf. A366358.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(4*k-1, k)*binomial(3*k-1, n-k)/(4*k-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(4*k-1,k) * binomial(3*k-1,n-k)/(4*k-1).

A366366 G.f. satisfies A(x) = (1 + x/A(x)^4)/(1 - x).

Original entry on oeis.org

1, 2, -6, 58, -574, 6402, -75878, 939290, -12000318, 157050178, -2094657926, 28368411194, -389079656446, 5393118559938, -75431624084838, 1063251390845338, -15088643098754942, 215396586102923138, -3091050571516120582, 44566089825496186170
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366364, A366365.
Cf. A366359.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k-1, k)*binomial(4*k-1, n-k)/(5*k-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k-1,k) * binomial(4*k-1,n-k)/(5*k-1).

A363816 G.f. satisfies A(x) = (1 + x/A(x))/(1 - x)^2.

Original entry on oeis.org

1, 3, 2, 8, -9, 62, -230, 1054, -4753, 22208, -105419, 508396, -2482284, 12248430, -60980860, 305955372, -1545397447, 7852100312, -40105277621, 205798130624, -1060467961487, 5485199090834, -28469067353663, 148220323891484, -773892318396664, 4051261817405034
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2023

Keywords

Links

Crossrefs

Partial sums of A366356.
Cf. A162477, A363818, A364629, A364630.
Cf. A363817, A366363.

Programs

  • Mathematica
    A363816[n_]:=(-1)^(n-1)Sum[Binomial[2k-1,k]Binomial[2(k-1),n-k]/(2k-1),{k,0,n}];Array[A363816,30,0] (* Paolo Xausa, Oct 20 2023 *)
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(2*(k-1), n-k)/(2*k-1));

Formula

G.f.: A(x) = -2*x / (1-sqrt(1+4*x*(1-x)^2)).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(2*(k-1),n-k)/(2*k-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

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.

A363817 G.f. satisfies A(x) = (1 + x/A(x))/(1 - x)^3.

Original entry on oeis.org

1, 4, 5, 15, -5, 111, -402, 2172, -10892, 57362, -305756, 1656560, -9083341, 50328219, -281324174, 1584578882, -8984740332, 51242962422, -293772467974, 1691974930794, -9785378133066, 56805049768410, -330880419984556, 1933299689139664, -11328101469158229
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2023

Keywords

Crossrefs

Cf. A366363, A363816.

Programs

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

Formula

G.f.: A(x) = -2*x / (1-sqrt(1+4*x*(1-x)^3)).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(3*(k-1),n-k)/(2*k-1).

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.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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