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.

A354729 E.g.f. A(x) satisfies A(x) = 1 + x * A(log(1+x)).

Original entry on oeis.org

1, 1, 2, 3, -4, -30, 234, 679, -35848, 305208, 6762360, -290545486, 2866197828, 186075548048, -10575881477630, 151622861284395, 14937532353298992, -1269964031741331704, 32904195657758601624, 2814524425307181390432, -395787864674458924551840
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2022

Keywords

Crossrefs

Cf. A048801, A354728, A354730.

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, 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 * A354728(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

Cf. A135750, A213357, A354574, A354728, A354730.

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).

A355096 E.g.f. A(x) satisfies A(x) = 1 + 2 * log(1+x) * A(log(1+x)).

Original entry on oeis.org

1, 2, 6, 16, -12, -492, 628, 63488, -408112, -20183928, 444216616, 9449212584, -679737200176, 2572902869080, 1276955484043864, -53294396490490656, -1891642613896659904, 314259171327032640928, -8590801196259162852288, -1381246455381881103425424
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2022

Keywords

Crossrefs

Cf. A354728, A355097.
Cf. A355104.

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*stirling(i, j, 1)*v[j])); v;

Formula

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

A355133 E.g.f. A(x) satisfies A(x) = 1 + 2 * log(1+x) * A(2 * log(1+x)).

Original entry on oeis.org

1, 2, 14, 292, 16836, 2517888, 927979616, 811623678304, 1639230314891936, 7494183556478948928, 76401967141928846136512, 1716972732272402536841957760, 84279193103775042893631925450624, 8968818994749615710061662692132983296
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2022

Keywords

Crossrefs

Cf. A354728, A355120, A355124.

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*stirling(i, j, 1)*v[j])); v;

Formula

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

A355097 E.g.f. A(x) satisfies A(x) = 1 + 3 * log(1+x) * A(log(1+x)).

Original entry on oeis.org

1, 3, 15, 87, 414, -333, -36207, -125415, 9692208, 65346480, -6686193834, -28979410023, 9399236141664, -74034590428035, -21601690453093869, 753341807730002715, 64208547156310265880, -6137571162315494165580, -131200755856066508312736
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2022

Keywords

Crossrefs

Cf. A354728, A355096.
Cf. A355105.

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*stirling(i, j, 1)*v[j])); v;

Formula

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

A355120 E.g.f. A(x) satisfies A(x) = 1 + log(1+x) * A(2 * log(1+x)).

Original entry on oeis.org

1, 1, 3, 26, 654, 45084, 7934924, 3381663872, 3365978050576, 7632454575648720, 38732162420625498608, 434139952882119137261024, 10640704036253473615712677216, 565765176687479152385624223741568, 64834956096893473256448986077914291328
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2022

Keywords

Crossrefs

Cf. A354728, A355084, A355096, A355133.

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)*stirling(i, j, 1)*v[j])); v;

Formula

E.g.f. A(x) satisfies: A(exp(x) - 1) = 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 1-6 of 6 results.

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