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 1-7 of 7 results.

A213357 E.g.f. satisfies A(x) = 1 + (exp(x) - 1) * A(exp(x) - 1).

Original entry on oeis.org

1, 1, 3, 16, 133, 1561, 24374, 485640, 11969843, 356348290, 12572687675, 517644938724, 24553141710156, 1327223189312606, 81005220402829714, 5537660009982114858, 421050946315817655785, 35387457515051683169307, 3269500807582223015227780
Offset: 0

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Author

Michael Somos, Jun 09 2012

Keywords

Examples

			1 + x + 3*x^2 + 16*x^3 + 133*x^4 + 1561*x^5 + 24374*x^6 + 485640*x^7 + ...
		

Crossrefs

Programs

  • Mathematica
    nmax=20; b = ConstantArray[0,nmax+1]; b[[1]]=1; Do[b[[n+1]] = Sum[k*b[[k]]*StirlingS2[n, k],{k,1,n}],{n,1,nmax-1}]; b (* Vaclav Kotesovec, Mar 12 2014 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = subst( 1 + x * A, x, exp( x + x * (A - A)) - 1)); n! * polcoeff( A, n))}
    
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*stirling(i, j, 2)*v[j])); v; \\ Seiichi Manyama, Jun 04 2022

Formula

a(n) = Sum_{k=1..n} k * a(k-1) * Stirling2(n, k) if n>0.
A048801(n) = n * a(n-1) = Sum_{k=1..n} a(k) * Stirling1(n, k) if n>0.

A354728 E.g.f. A(x) satisfies A(x) = 1 + log(1+x) * A(log(1+x)).

Original entry on oeis.org

1, 1, 1, -1, -6, 39, 97, -4481, 33912, 676236, -26413226, 238849819, 14313503696, -755420105545, 10108190752293, 933595772081187, -74703766573019512, 1828010869875477868, 148132864489851652128, -19789393233722946227592, 910780967051245532791008
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2022

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*stirling(i, j, 1)*v[j])); v;

Formula

E.g.f. A(x) satisfies: A(exp(x) - 1) = 1 + x*A(x).
a(0) = 1; a(n) = Sum_{k=1..n} k * Stirling1(n,k) * a(k-1).

A354730 E.g.f. A(x) satisfies: A(x) = 1 + x * A(-log(1-x)).

Original entry on oeis.org

1, 1, 2, 9, 68, 750, 11214, 216559, 5217176, 152742528, 5324034480, 217322508194, 10248159667140, 551968543756448, 33627829222316770, 2298114390067518705, 174893648144384932176, 14727538317383970615352, 1364522959678851181512504
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2022

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*sum(j=0, i-1, abs(stirling(i-1, j, 1))*v[j+1])); v;

Formula

a(0) = 1; a(n) = n * Sum_{k=0..n-1} |Stirling1(n-1,k)| * a(k).
a(n) = n * A135750(n-1) for n>0.

A353177 E.g.f. A(x) satisfies A(x) = 1 + (1 - exp(-x)) * A(1 - exp(-x)).

Original entry on oeis.org

1, 1, 1, -2, -13, 61, 612, -8924, -41991, 2821876, -22689807, -1196339088, 45175812442, 10968806278, -63633205318330, 2495113782094766, 31372553334367367, -8832192422722410665, 421480840601004167822, 9536361803340658184343
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2022

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (-1)^(i-j)*j*stirling(i, j, 2)*v[j])); v;

Formula

E.g.f. A(x) satisfies: A(-log(1-x)) = 1 + x*A(x).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(n-k) * k * Stirling2(n,k) * a(k-1).

A355098 E.g.f. A(x) satisfies A(x) = 1 - 2 * log(1-x) * A(-log(1-x)).

Original entry on oeis.org

1, 2, 10, 88, 1164, 21228, 505108, 15088400, 549924048, 23922798360, 1220592387496, 72008007861128, 4853864641010384, 370112914857814360, 31651011896528812776, 3013092750843813488640, 317232128940068230592960, 36726669357239166496674080
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2022

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=2*sum(j=1, i, j*abs(stirling(i, j, 1))*v[j])); v;

Formula

E.g.f. A(x) satisfies: A(1 - exp(-x)) = 1 + 2*x*A(x).
a(0) = 1; a(n) = 2 * Sum_{k=1..n} k * |Stirling1(n,k)| * a(k-1).

A355099 E.g.f. A(x) satisfies A(x) = 1 - 3 * log(1-x) * A(-log(1-x)).

Original entry on oeis.org

1, 3, 21, 249, 4338, 102537, 3123513, 118277037, 5420074248, 294405725628, 18643757033286, 1357970251340601, 112491520189940304, 10497256870300840845, 1094461858289007808209, 126592088471657042694381, 16143127318109911141849896, 2257107645258692949884188932
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2022

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, j*abs(stirling(i, j, 1))*v[j])); v;

Formula

E.g.f. A(x) satisfies: A(1 - exp(-x)) = 1 + 3*x*A(x).
a(0) = 1; a(n) = 3 * Sum_{k=1..n} k * |Stirling1(n,k)| * a(k-1).

A355121 E.g.f. A(x) satisfies A(x) = 1 - log(1-x) * A(-2 * log(1-x)).

Original entry on oeis.org

1, 1, 5, 74, 2778, 248244, 51212444, 23984832416, 25218677193200, 59000757443457072, 304720138059811544048, 3449059394896458379058208, 84991203371449537414272981856, 4532232538284485346856696552505728, 520204832574009673696495358635072488576
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2022

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*2^(j-1)*abs(stirling(i, j, 1))*v[j])); v;

Formula

E.g.f. A(x) satisfies: A(1 - exp(-x)) = 1 + x*A(2*x).
a(0) = 1; a(n) = Sum_{k=1..n} k * 2^(k-1) * |Stirling1(n,k)| * a(k-1).
Showing 1-7 of 7 results.