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 1-4 of 4 results.

A302547 Expansion of e.g.f. -log(1 - log(1 + x))/(1 - log(1 + x)).

Original entry on oeis.org

0, 1, 2, 4, 11, 33, 131, 516, 2810, 12934, 97870, 447940, 5308112, 16394116, 450505844, -315178912, 60774618672, -394330113648, 12662225550288, -157622647720032, 3766647294946944, -64679214198647520, 1475157821754785184, -30431206030329719424, 719032203373502252160
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 4*x^3/3! + 11*x^4/4! + 33*x^5/5! + 131*x^6/6! + ...

Links

Crossrefs

Cf. A000254, A001008, A002805, A006252, A073596, A089064, A222058, A300490, A302548.

Programs

  • Maple
    H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> add(Stirling1(n, k)*H(k)*k!, k=1..n):
seq(a(n), n=0..27);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 24; CoefficientList[Series[-Log[1 - Log[1 + x]]/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 24}]

Formula

a(n) = Sum_{k=1..n} Stirling1(n,k)*H(k)*k!, where H(k) is the k-th harmonic number.

A302548 Expansion of e.g.f. -log(1 + log(1 - x))/(1 + log(1 - x)).

Original entry on oeis.org

0, 1, 4, 22, 155, 1333, 13541, 158688, 2107682, 31291894, 513590170, 9234669420, 180534475832, 3812852144788, 86517295628188, 2099170738243328, 54233876338638192, 1486517654443664016, 43084555863325589232, 1316588795487600071904, 42306543064537291007424, 1426115146736949130634400
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 22*x^3/3! + 155*x^4/4! + 1333*x^5/5! + 13541*x^6/6! + ...

Links

Crossrefs

Cf. A000254, A001008, A002805, A003713, A007840, A073596, A222058, A300490, A302547.

Programs

  • Maple
    H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*H(k)*k!, k=1..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[-Log[1 + Log[1 - x]]/(1 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 21}]

Formula

a(n) = Sum_{k=1..n} |Stirling1(n,k)|*H(k)*k!, where H(k) is the k-th harmonic number.
a(n) ~ sqrt(2*Pi) * log(n) * n^(n + 1/2) / (exp(1)-1)^(n+1). - Vaclav Kotesovec, Jun 23 2018

A269954 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 5, 3, 1, 0, 9, 20, 17, 6, 1, 0, 44, 109, 100, 45, 10, 1, 0, 265, 689, 694, 355, 100, 15, 1, 0, 1854, 5053, 5453, 3094, 1015, 196, 21, 1, 0, 14833, 42048, 48082, 29596, 10899, 2492, 350, 28, 1
Offset: 0

Views

Author

Peter Luschny, Apr 12 2016

Keywords

Examples

			Triangle starts:
  1;
  0,   1;
  0,   0,   1;
  0,   1,   1,   1;
  0,   2,   5,   3,   1;
  0,   9,  20,  17,   6,   1;
  0,  44, 109, 100,  45,  10,  1;
  0, 265, 689, 694, 355, 100, 15, 1;

Links

Crossrefs

A000255 (row sums), A000217 (diag. n,n-1), A133252 (diag. n,n-2).
Columns k=0..4 give A000007, A000166(n-1), A300490(n-1), A381067(n-1), A381068(n-1).
Cf. A269951, A269953.

Programs

  • Maple
    A269954 := (n, k) -> add(binomial(-j, -n)*abs(Stirling1(j, k)), j=0..n):
seq(seq(A269954(n, k), k=0..n), n=0..9);
  • Mathematica
    Flatten[Table[Sum[Binomial[-j,-n] Abs[StirlingS1[j,k]],{j,0,n}], {n,0,9},{k,0,n}]]
  • PARI
    T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n-1, n-j)*abs(stirling(j, k)));
for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))); \\ Seiichi Manyama, Feb 13 2025

A381064 Expansion of e.g.f. log(1-x)^2 * exp(-x) / 2.

Original entry on oeis.org

0, 0, 1, 0, 5, 15, 94, 595, 4458, 37590, 354051, 3682646, 41935695, 518954293, 6935360496, 99553094537, 1527716784020, 24959724735564, 432572721886437, 7926615468800172, 153129657663788761, 3110514839038091643, 66278515188844197218, 1478222957082474301887
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2025

Keywords

Crossrefs

Column k=2 of A269953.
Cf. A300490, A381021.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*abs(stirling(k, 2, 1)));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * |Stirling1(k,2)|.
Showing 1-4 of 4 results.

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