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.

A166588 Partial sums of A097331; binomial transform of A166587.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 5, 10, 10, 24, 24, 66, 66, 198, 198, 627, 627, 2057, 2057, 6919, 6919, 23715, 23715, 82501, 82501, 290513, 290513, 1033413, 1033413, 3707853, 3707853, 13402698, 13402698, 48760368, 48760368, 178405158, 178405158, 656043858
Offset: 0

Views

Author

Paul Barry, Oct 17 2009

Keywords

Comments

Hankel transform is A131713. The Hankel transform of the sequence 1,1,2,2,... is A128017(n+3). A155587 doubled.

Links

Programs

  • Mathematica
    CoefficientList[Series[(1+2*x-Sqrt[1-4*x^2])/(2*x*(1-x)), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Formula

G.f.: (1+2x-sqrt(1-4x^2))/(2x(1-x))=((1+x^2*c(x^2))/(1-x)-1)/x, c(x) the g.f. of A000108.
a(n) = Sum_{k=0..n} C(n,k)*A166587(k).
Conjecture: (-n-1)*a(n) + (n+1)*a(n-1) + 4*(n-2)*a(n-2) + 4*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012
a(n) ~ 2^(n+1/2) * (3-(-1)^n) / (3 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 08 2014

A168051 Expansion of (1+x+sqrt(1-2x-3x^2))/2.

Original entry on oeis.org

1, 0, -1, -1, -2, -4, -9, -21, -51, -127, -323, -835, -2188, -5798, -15511, -41835, -113634, -310572, -853467, -2356779, -6536382, -18199284, -50852019, -142547559, -400763223, -1129760415, -3192727797, -9043402501, -25669818476
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Comments

A signed variant of the Motzkin numbers A001006. Hankel transform is A168052.

Examples

			G.f. = 1 - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 21*x^7 - 51*x^8 - 127*x^9 + ...

Links

Crossrefs

Cf. A168049, A166587, A167022.
Cf. A001006, A005043.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1 + x + Sqrt[1 - 2 x - 3 x^2]) / 2, {x, 0, n}] (* Michael Somos, Jan 25 2014 *)
  • PARI
    {a(n) = polcoeff( (1 + x + sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2, n)}; /* Michael Somos, Jan 25 2014 */

Formula

D-finite with recurrence: n*a(n) -(2n-3)*a(n-1) -3*(n-3)*a(n-2)=0 if n>2. - R. J. Mathar, Dec 20 2011 [Edited by Michael Somos, Jan 25 2014]
0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 25 2014
G.f.: 1 + x - (x + x^2) / (1 + x - (x + x^2) / (1 + x - ...)). - Michael Somos, Mar 27 2014
Convolution inverse of A005043. - Michael Somos, Mar 27 2014
a(n) ~ -3^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 05 2018
From Gennady Eremin, Feb 25 2021: (Start)
For n > 1, a(n) = A167022(n) / 2.
G.f.: (1 + x + A(x)) / 2, where A(x) is the g.f. of A167022. (End)

A372013 G.f. A(x) satisfies A(x) = 1/( 1 + x*(1 - 9*x*A(x))^(1/3) ).

Original entry on oeis.org

1, -1, 4, -1, 46, 149, 1351, 8441, 63499, 462752, 3514807, 26923478, 209566927, 1647633779, 13079663527, 104649229988, 843120766711, 6833665175513, 55683581174641, 455878084448132, 3748025535972448, 30931714278955736, 256150668109462507
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2024

Keywords

Crossrefs

Cf. A166587, A362155, A372012.

Programs

  • PARI
    a(n) = (-1)^n*sum(k=0, n, 9^(n-k)*binomial(n, k)*binomial(k/3, n-k)/(n-k+1));

Formula

a(n) = (-1)^n * Sum_{k=0..n} 9^(n-k) * binomial(n,k) * binomial(k/3,n-k)/(n-k+1).

A372012 G.f. A(x) satisfies A(x) = 1/( 1 + x*(1 - 4*x*A(x))^(1/2) ).

Original entry on oeis.org

1, -1, 3, -5, 17, -31, 119, -211, 937, -1483, 8015, -10187, 73369, -62193, 713907, -234857, 7358657, 1881661, 80117735, 69295469, 917837521, 1334044075, 11006114883, 21830065899, 137275956089, 333858963899, 1769128762419, 4940496514271, 23409778504937
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2024

Keywords

Crossrefs

Cf. A166587, A362154, A372013.

Programs

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

Formula

a(n) = (-1)^n * Sum_{k=0..n} 4^(n-k) * binomial(n,k) * binomial(k/2,n-k)/(n-k+1).

A378783 Triangular array T(n,k) read by rows: T(n, k) = c_k(n+1). The sequence c_k(m) has the ordinary generating function C_k(x) which satisfies C_k(x) = 1/(1+C_k(x)*Sum_{t=0..k} x^(t+1)).

Original entry on oeis.org

-1, 2, 1, -5, -1, -2, 14, 1, 5, 4, -42, -1, -12, -8, -9, 132, 1, 29, 18, 22, 21, -429, -1, -73, -43, -54, -50, -51, 1430, 1, 190, 105, 135, 124, 128, 127, -4862, -1, -505, -262, -345, -315, -326, -322, -323, 16796, 1, 1363, 666, 896, 813, 843, 832, 836, 835
Offset: 0

Views

Author

Thomas Scheuerle, Dec 07 2024

Keywords

Examples

			Triangle begins:
  [0]    -1
  [1]     2,  1
  [2]    -5, -1,   -2
  [3]    14,  1,    5,    4
  [4]   -42, -1,  -12,   -8,   -9
  [5]   132,  1,   29,   18,   22,   21
  [6]  -429, -1,  -73,  -43,  -54,  -50,  -51
  [7]  1430,  1,  190,  105,  135,  124,  128,  127
  [8] -4862, -1, -505, -262, -345, -315, -326, -322, -323
.

Crossrefs

Cf. A001006, A000108, A152171, A166587, A168491, A378816.

Programs

  • Mathematica
    T[n_,k_]:=SeriesCoefficient[(2 / (Sqrt[1+4*Sum[x^(t+1),{t,0,k}]] + 1) - 1)/x,{x,0,n}];Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)
  • PARI
    column(n, max_n) = { my(s = 1,x = 'x,cu); for(k = 0, max_n-1, cu = cu+polcoeff(1/s+O(x^(k+1)), k, x); cu = cu-polcoeff(1/s+O(x^(k+1)), k-1-n, x); s = s+cu*x^(k+1)); Vec(1/s+O(x^max_n)) };
T(n, k) = column(k, n+2)[n+2]
T(n, k) = polcoeff(2 / (sqrt(1+4*x*sum(t=0, k, x^t)) + 1) + O(x^(n+2)), n+1, x)

Formula

G.f. column k: (2 / (sqrt(1+4*Sum_{t=0..k}x^(t+1)) + 1) - 1)/x.
T(n, 0) = (-1)^(n+1)*Catalan(n+1) = A168491(n+1).
T(n, 2) = (-1)^(n+1)*A152171(n+1).
T(n, n) = (-1)^(n+1)*A001006(n) = -A166587(n+1).
A378816(n) = Limit_{k->oo} (T(k, k-n) - T(k, k-n-1)).
Showing 1-5 of 5 results.

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