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 1-6 of 6 results.

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

A211824 G.f. satisfies: A(x) = 1 + x*( d/dx x*A(x) )^3.

Original entry on oeis.org

1, 1, 6, 66, 1016, 19596, 447312, 11686008, 341966304, 11044539840, 389511815136, 14879686213728, 611795661826176, 26934556130346880, 1264203675152355840, 63023836596988857216, 3326204117173583906304, 185302040367551696870400, 10868134346437165639956480
Offset: 0

Views

Author

Paul D. Hanna, Apr 21 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 66*x^3 + 1016*x^4 + 19596*x^5 + 447312*x^6 +...
Related expansions:
d/dx x*A(x) = 1 + 2*x + 18*x^2 + 264*x^3 + 5080*x^4 + 117576*x^5 +...
A'(x) = 1 + 12*x + 198*x^2 + 4064*x^3 + 97980*x^4 + 2683872*x^5 +...

Links

Crossrefs

Cf. A112915, A211825, A211826.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*deriv(x*A)^3);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1 + x*(A(x) + x*A'(x))^3.
a(n) ~ c * 3^n * n! * n^(4/3), where c = 0.1005380575409567... - Vaclav Kotesovec, Aug 24 2017

A211825 G.f. satisfies: A(x) = 1 + x*( d/dx x*A(x) )^4.

Original entry on oeis.org

1, 1, 8, 120, 2528, 66704, 2080128, 74115840, 2952926720, 129637843968, 6205231472640, 321275171444736, 17880710254829568, 1064356462925701120, 67476012302577762304, 4539384115900126199808, 323034928746773883518976, 24248087962137553507450880
Offset: 0

Views

Author

Paul D. Hanna, Apr 21 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 120*x^3 + 2528*x^4 + 66704*x^5 + 2080128*x^6 +...
Related expansions:
d/dx x*A(x) = 1 + 2*x + 24*x^2 + 480*x^3 + 12640*x^4 + 400224*x^5 +...
A'(x) = 1 + 16*x + 360*x^2 + 10112*x^3 + 333520*x^4 + 12480768*x^5 +...

Links

Crossrefs

Cf. A112915, A211824, A211826.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*deriv(x*A)^4);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1 + x*(A(x) + x*A'(x))^4.
a(n) ~ c * 4^n * n! * n^(3/2), where c = 0.06185263969861377609335... - Vaclav Kotesovec, Aug 24 2017

A211826 G.f. satisfies: A(x) = 1 + x*( d/dx x*A(x) )^5.

Original entry on oeis.org

1, 1, 10, 190, 5080, 170080, 6724432, 303476320, 15300084160, 849174449680, 51341667458240, 3354970165353120, 235493617889171200, 17667618435092524160, 1410845692308772162560, 119491232651437498097920, 10700209630623386429434880, 1010278582501924072528588800
Offset: 0

Views

Author

Paul D. Hanna, Apr 21 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 10*x^2 + 190*x^3 + 5080*x^4 + 170080*x^5 +...
Related expansions:
d/dx x*A(x) = 1 + 2*x + 30*x^2 + 760*x^3 + 25400*x^4 + 1020480*x^5 +...
A'(x) = 1 + 20*x + 570*x^2 + 20320*x^3 + 850400*x^4 + 40346592*x^5 +...

Links

Crossrefs

Cf. A112915, A211824, A211825.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*deriv(x*A)^5);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1 + x*(A(x) + x*A'(x))^5.
a(n) ~ c * 5^n * n^(8/5) * n!, where c = 0.04375376183367762... - Vaclav Kotesovec, Aug 24 2017

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])}

A218222 G.f. A(x) satisfies: A(x) = x + x*[d/dx A(x)^2].

Original entry on oeis.org

1, 2, 12, 112, 1360, 19872, 335104, 6359040, 133560576, 3069007360, 76493880320, 2054400577536, 59136549994496, 1816392567062528, 59305340822814720, 2051451257317490688, 74958908119819812864, 2885480280276224311296, 116731741304854533111808
Offset: 1

Views

Author

Paul D. Hanna, Jan 31 2013

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 12*x^3 + 112*x^4 + 1360*x^5 + 19872*x^6 +...
Related series:
A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 272*x^5 + 3312*x^6 + 47872*x^7 + 794880*x^8 + 14840064*x^9 +...+ A112915(n-1)*x^n +...
d/dx A(x)^2 = 2*x + 12*x^2 + 112*x^3 + 1360*x^4 + 19872*x^5 +...

Links

Crossrefs

Cf. A000699, A112915, A185183.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1,
      n*add(a(i)*a(n-i), i=1..n-1))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 05 2020
  • Mathematica
    a[n_] := a[n] = If[n<2, 1, n*Sum[a[i]*a[n-i], {i, 1, n-1}]];
Array[a, 20] (* Jean-François Alcover, Dec 18 2020, after Maple *)
  • PARI
    {a(n)=local(A=x+x^2); for(i=1, n, A=x+x*deriv(A^2+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = 2^(n-1)*A088716(n-1) for n>=1, where g.f. F(x) of A088716 satisfies: F(x) = 1 + x*F(x)*d/dx[x*F(x)].
a(n) = n*A112915(n-1) for n>=1.
G.f.: x*d/dx x*G(x), where g.f. G(x) of A112915 satisfies: G(x) = 1 + x*(d/dx[x*G(x)])^2.
a(n) ~ c * n * 2^(n-1) * n!, where c = A238223 = 0.21795078944715106549... - Vaclav Kotesovec, Aug 24 2017
Showing 1-6 of 6 results.

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