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.

A112912 Column 1 of triangle A112911.

Original entry on oeis.org

1, 2, 8, 44, 298, 2358, 21154, 211100, 2313030, 27566654, 354806116, 4903884712, 72444584732, 1139381007880, 19012236634968, 335560664081388, 6247230709277958, 122375974186267566, 2516528486416495240
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Crossrefs

Cf. A088716, A112911.

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=2,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1]);));A=B);return((A^-1)[n+2,2])}

Formula

a(n) = Sum_{k=0..n-1} (n-k+1)*A088716(k)*a(n-k-1) for n>0 with a(0)=1.

A112913 Column 2 of triangle A112911.

Original entry on oeis.org

1, 3, 15, 96, 729, 6327, 61380, 655944, 7642557, 96338925, 1306003803, 18947038404, 292956751236, 4810423131828, 83620543389804, 1534493081643552, 29649997606132197, 601828286370137865, 12804746026265284149
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Crossrefs

Cf. A112911 (triangle), A088716 (column 0), A112912 (column 1), A112914 (column 3).

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=2,n+3,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1]);));A=B);return((A^-1)[n+3,3])}

Formula

a(n) = Sum_{k=0..n-1} (n-k+2)*A088716(k)*a(n-k-1) for n>0 with a(0)=1.

A112914 Column 3 of triangle A112911.

Original entry on oeis.org

1, 4, 24, 176, 1492, 14220, 149812, 1724760, 21519108, 289101836, 4160750192, 63873461064, 1042061108096, 18008563271632, 328712410879632, 6320649807989400, 127724091041379492, 2706397280528315148
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Crossrefs

Cf. A112911 (triangle), A088716 (column 0), A112912 (column 1), A112913 (column 2).

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=2,n+4,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1]);));A=B);return((A^-1)[n+4,4])}

Formula

a(n) = Sum_{k=0..n-1} (n-k+3)*A088716(k)*a(n-k-1) for n>0 with a(0)=1.

A088716 G.f. satisfies: A(x) = 1 + x*A(x)*d/dx[x*A(x)] = 1 + x*A(x)^2 + x^2*A(x)*A'(x).

Original entry on oeis.org

1, 1, 3, 14, 85, 621, 5236, 49680, 521721, 5994155, 74701055, 1003125282, 14437634276, 221727608284, 3619710743580, 62605324014816, 1143782167355649, 22014467470369143, 445296254367273457, 9444925598142843970
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003

Keywords

Links

Crossrefs

Cf. A112916 (A^2), A112911, A112912, A112913, A112914.
Cf. A088715, A182962, A112915, A218222, A238223.
Cf. A300736, A300987, A300989, A300991, A300993.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(j)*a(n-j-1)*(j+1), j=0..n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 10 2017
  • Mathematica
    a=ConstantArray[0,21]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = Sum[k*a[[n-k+1]]*a[[k]],{k,1,n}],{n,2,20}]; a (* Vaclav Kotesovec, Feb 21 2014 *)
m = 20; A[_] = 0;
Do[A[x_] = 1 + x A[x]^2 + x^2 A[x] A'[x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Feb 18 2020 *)
a[1]:=1; a[2]:=1; a[n_]:=a[n]=n/2 Sum[a[k] a[n-k], {k,1,n-1}];
Map[a,Range[20]] (* Oliver Seipel, Nov 03 2024 ,after Schröder 1870 *)
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,(k+1)*a(k)*a(n-k-1)))
  • PARI
    {a(n)=local(G=1+x);for(i=1,n,G=exp(x/(1 - x*deriv(G)/G+x*O(x^n))));polcoeff(log(G)/x,n)} \\ Paul D. Hanna, Jan 01 2011

Formula

a(n) = Sum_{k=1..n} k*a(k-1)*a(n-k) for n>=1 with a(0)=1.
Forms column 0 of triangle T=A112911, where the matrix inverse satisfies [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0.
Self-convolution is A112916, where a(n) = (n+1)/2*A112916(n-1) for n>0.
G.f.: A(x) = serreverse(x/f(x))/x where f(x) is the g.f. of A088715.
O.g.f.: A(x) = log(G(x))/x where G(x) is the e.g.f. of A182962 given by G(x) = exp( x/(1 - x*G'(x)/G(x)) ). [Paul D. Hanna, Jan 01 2011]
O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) / A(x) = 0 for n>0. - Paul D. Hanna, May 25 2018
O.g.f. A(x) satisfies [x^n] exp( n * x*A(x) ) * (1 - n*x) = 0 for n>0. - Paul D. Hanna, Jul 24 2019
From Paul D. Hanna, Jul 20 2018 (Start):
O.g.f. A(x) satisfies:
* [x^n] exp(-n * x*A(x)) * (2 - 1/A(x)) = 0 for n >= 1.
* [x^n] exp(-n^2 * x*A(x)) * (n + 1 - n/A(x)) = 0 for n >= 1.
* [x^n] exp(-n^(p+1) * x*A(x)) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0. (End)
a(n) ~ c * n! * n^2, where c = 0.21795078944715106549282282244231982088... (see A238223). - Vaclav Kotesovec, Feb 21 2014

A112916 Self-convolution of A088716, where a(n) = 2*A088716(n+1)/(n+2) for n>=0.

Original entry on oeis.org

1, 2, 7, 34, 207, 1496, 12420, 115938, 1198831, 13582010, 167187547, 2221174504, 31675372612, 482628099144, 7825665501852, 134562607924194, 2446051941152127, 46873289933397206, 944492559814284397
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Crossrefs

Cf. A088716, A112915, A112911.

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1]);));A=B); return(Vec(Ser(vector(n+1,i,(A^-1)[i,1]))^2)[n+1])}
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.