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.

A102221 Column 0 of triangular matrix A102220, which equals [2*I - A008459]^(-1).

Original entry on oeis.org

1, 1, 5, 55, 1077, 32951, 1451723, 87054773, 6818444405, 675900963271, 82717196780955, 12248810651651333, 2158585005685222491, 446445657799551807541, 107087164031952038620481, 29487141797206760561836055, 9238158011747884080353808245
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2004

Keywords

Comments

a(n) is the number of ways to form an ordered pair of n-permutations and then choose a subset of its common descent set. Cf. A192721. - Geoffrey Critzer, Apr 29 2023

Links

Crossrefs

Cf. A000275, A008459, A102220, A102222, A192721.
Row sums of A192722.
Column k=2 of A326322.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n, i)/i!, i=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 11 2016
  • Mathematica
    Rest[CoefficientList[Series[1/(2-BesselJ[0, 2*I*Sqrt[x]]), {x, 0, 20}], x] * Range[0, 20]!^2] (* Vaclav Kotesovec, Mar 02 2014 *)
m = 20; CoefficientList[1/(2 - BesselI[0, 2 Sqrt[x]]) + O[x]^m, x] Range[0, m - 1]!^2 (* Jean-François Alcover, Jun 11 2019, after Vladeta Jovovic *)
b[n_] := b[n] = If[n==0, 1, Sum[b[n-i] Binomial[n, i]/i!, {i, 1, n}]];
a[n_] := b[n] n!;
a /@ Range[0, 20] (* Jean-François Alcover, Dec 03 2020, after Alois P. Heinz *)
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)^2*a(k)))
  • Sage
    L = taylor(1/(1-x*hypergeometric((1,),(2,2),x)),x,0,14).list()
[factorial(i)^2*c for (i,c) in enumerate(L)] # Peter Luschny, Jul 28 2015

Formula

a(n) = Sum_{k=0..n-1} C(n, k)^2*a(k) for n>0, with a(0)=1.
a(n) = A102220(n+k, k)/C(n+k, k)^2 for k>=0.
Sum_{n>=0} a(n)*x^n/n!^2 = 1/(2-BesselI(0,2*sqrt(x))). - Vladeta Jovovic, Jul 17 2006
a(n) ~ c * (n!)^2 / r^n, where r = 0.81712266563155429332453954757369795... is the root of the equation BesselJ(0, 2*I*sqrt(x))=2, and c = 0.833570458821600548332410448635741072476086046022299770387... = 1/(sqrt(r) * BesselI(1, 2*sqrt(r))). - Vaclav Kotesovec, Mar 02 2014, updated Apr 01 2018
From Geoffrey Critzer, Apr 29 2023: (Start)
Sum_{n>=0} a(n)*z^n/(n!)^2 = 1/(2-E(z)) where E(z) = Sum_{n>=0} z^n/(n!)^2.
a(n) = Sum_{k=0..n-1} A192721(n,k)*2^k. (End)

Extensions

Content moved from A192723 to this sequence by Alois P. Heinz, Sep 11 2019

A102222 Logarithm of triangular matrix A102220, which equals [2*I - A008459]^(-1).

Original entry on oeis.org

0, 1, 0, 3, 4, 0, 22, 27, 9, 0, 323, 352, 108, 16, 0, 7906, 8075, 2200, 300, 25, 0, 290262, 284616, 72675, 8800, 675, 36, 0, 14919430, 14222838, 3486546, 395675, 26950, 1323, 49, 0, 1022475715, 954843520, 227565408, 24793216, 1582700, 68992, 2352, 64, 0
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2004

Keywords

Comments

Column 0 forms A102223.

Examples

			Rows begin:
[0],
[1,0],
[3,4,0],
[22,27,9,0],
[323,352,108,16,0],
[7906,8075,2200,300,25,0],
[290262,284616,72675,8800,675,36,0],...
which equals the term-by-term product of column 0
with the squared binomial coefficients (A008459):
[(0)1^2],
[(1)1^2,(0)1^2],
[(3)1^2,(1)2^2,(0)1^2],
[(22)1^2,(3)3^2,(1)3^2,(0)1^2],
[(323)1^2,(22)4^2,(3)6^2,(1)4^2,(0)1^2],...

Crossrefs

Cf. A008459, A102220, A102223.

Programs

  • PARI
    {T(n,k)=if(n

Formula

T(n, k) = C(n, k)^2*A102223(n-k). T(n, 0) = A102223(n). T(n, n) = 0 for n>=0. [A102222] = Sum_{m=1..inf} [A008459 - I]^m/m.

A102224 Column 0 of the matrix square of A102220, which equals the lower triangular matrix: [2*I - A008459]^(-1).

Original entry on oeis.org

1, 2, 14, 200, 4814, 174752, 8909168, 606818060, 53211837134, 5838211285616, 783434682568664, 126221710572107900, 24043148814317769584, 5344827109234104188348, 1371307353540074156012828
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2004

Keywords

Comments

A102221 is column 0 of A102220.
Triangle A008459 consists of the squared binomial coefficients.

Examples

			Given A102221 = [1,1,5,55,1077,32951,1451723,87054773,...], then this sequence results from a type of self-convolution of A102221:
a(2) = 14 = 1^2*1*5 + 2^2*1*1 + 1^2*5*1,
a(3) = 200 = 1^2*1*55 + 3^2*1*5 + 3^2*5*1 + 1^2*55*1.

Crossrefs

Cf. A102220, A102221, A008459.

Programs

  • PARI
    {a(n)=(matrix(n+1,n+1,i,j,if(i==j,2,0)-binomial(i-1,j-1)^2)^-2)[n+1,1]}

Formula

a(n) = Sum_{k=0..n} C(n, k)^2*A102221(k)*A102221(n-k).
Sum_{n>=0} a(n)*x^n/n!^2 = 1/(2-BesselI(0,2*sqrt(x)))^2. - Vladeta Jovovic, Jul 17 2006

A102223 Column 0 of triangular matrix A102222, which equals -log[2*I - A008459].

Original entry on oeis.org

0, 1, 3, 22, 323, 7906, 290262, 14919430, 1022475715, 90094491994, 9923239949978, 1335853771297750, 215797095378591542, 41198645313603207990, 9176288655853717238830, 2358300288047799986966722
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2004

Keywords

Comments

Triangle A008459 consists of squared binomial coefficients.

Examples

			a(2) = 3 = 1 + (1*0*0 + 4*1*1)/2,
a(3) = 22 = 1 + (1*0*0 + 9*1*1 + 9*2*3)/3,
a(4) = 323 = 1 + (1*0*0 + 16*1*1 + 36*2*3 + 16*3*22)/4,
a(5) = 7906 = 1 + (1*0*0 + 25*1*1 + 100*2*3 + 100*3*22 + 25*4*323)/5.

Crossrefs

Cf. A008459, A102220, A102222.

Programs

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

Formula

a(n) = 1 + (1/n)*Sum_{k=0..n-1} C(n, k)^2*k*a(k) for n>0, with a(0)=0.
Sum_{n>=0} a(n)*x^n/n!^2 = -log(2-BesselI(0,2*sqrt(x))). - Vladeta Jovovic, Jul 16 2006

A335946 a(n) = 1 + Sum_{k=0..n-1} binomial(n,k)^2 * a(k).

Original entry on oeis.org

1, 2, 10, 110, 2154, 65902, 2903446, 174109546, 13636888810, 1351801926542, 165434393561910, 24497621303302666, 4317170011370444982, 892891315599103615082, 214174328063904077240962, 58974283594413521123672110, 18476316023495768160707616490
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 01 2020

Keywords

Links

Crossrefs

Row sums of A102220.
Cf. A000629, A102221.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(2 - BesselI[0, 2 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (2 - BesselI(0,2*sqrt(x))).
a(n) = 2 * A102221(n) for n > 0.
Showing 1-5 of 5 results.

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