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.

A354263 Expansion of e.g.f. 1/(1 + 3 * log(1-x)).

Original entry on oeis.org

1, 3, 21, 222, 3132, 55242, 1169262, 28873800, 814870584, 25871762016, 912684973968, 35416732159872, 1499286521185776, 68757945743286576, 3395829155786528976, 179693346163010491008, 10142543588881013369856, 608262031900883147262336
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Column k=3 of A320079.
Cf. A335531.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))))
  • 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-1)!*binomial(i, j)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n, 3^k*k!*abs(stirling(n, k, 1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).
a(n) = Sum_{k=0..n} 3^k * k! * |Stirling1(n, k)|.
a(n) ~ n! * exp(n/3) / (3 * (exp(1/3) - 1)^(n+1)). - Vaclav Kotesovec, Jun 04 2022

A320080 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - k*log(1 + x)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 2, 0, 1, 4, 15, 28, 4, 0, 1, 5, 28, 114, 172, 14, 0, 1, 6, 45, 296, 1152, 1328, 38, 0, 1, 7, 66, 610, 4168, 14562, 12272, 216, 0, 1, 8, 91, 1092, 11020, 73376, 220842, 132480, 600, 0, 1, 9, 120, 1778, 24084, 248870, 1550048, 3907656, 1633344, 6240, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 05 2018

Keywords

Examples

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k - 1)*x^2/2! + 2*k*(3*k^2 - 3*k + 1)*x^3/3! + 2*k*(12*k^3 - 18*k^2 + 11*k - 3)*x^4/4! + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   1,     6,     15,     28,      45,  ...
  0,   2,    28,    114,    296,     610,  ...
  0,   4,   172,   1152,   4168,   11020,  ...
  0,  14,  1328,  14562,  73376,  248870,  ...

Crossrefs

Columns k=0..5 give A000007, A006252, A088501, A335531, A354147, A365604.
Main diagonal gives A317172.
Cf. A048594, A048994, A094416, A320079, A334369.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[1/(1 - k Log[1 + x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: 1/(1 - k*log(1 + x)).
A(n,k) = Sum_{j=0..n} Stirling1(n,j)*j!*k^j.
A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (-1)^(j-1) * (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022

A365588 Expansion of e.g.f. 1 / (1 + 5 * log(1-x)).

Original entry on oeis.org

1, 5, 55, 910, 20080, 553870, 18333050, 707959800, 31244562600, 1551289408800, 85579293493200, 5193226343508000, 343790892166398000, 24655487205067386000, 1904221630155352038000, 157574022827034258192000, 13908505761692419540320000
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2023

Keywords

Links

Crossrefs

Column k=5 of A320079.
Cf. A346987, A365585, A365586, A365587.
Cf. A094418.

Programs

  • Mathematica
    a[n_] := Sum[5^k * k! * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, n, 5^k*k!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} 5^k * k! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 5 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (5 * exp(4*n/5) * (exp(1/5) - 1)^(n+1)). - Vaclav Kotesovec, Nov 11 2023

A317171 a(n) = n! * [x^n] 1/(1 + n*log(1 - x)).

Original entry on oeis.org

1, 1, 10, 222, 8824, 553870, 50545008, 6328330344, 1041597412224, 218138133235680, 56650689388344000, 17868469522986145536, 6728682216722958185472, 2981868816113406609186576, 1536217706761623823662025728, 910442461680276910819097616000, 615053979239579281793375485526016
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 23 2018

Keywords

Crossrefs

Cf. A007840, A088500, A094420, A242817, A317172.
Main diagonal of A320079.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 + n Log[1 - x]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] n^k k!, {k, n}], {n, 16}]]

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*n^k*k!.
a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n - 1/2). - Vaclav Kotesovec, Jul 23 2018

A354264 Expansion of e.g.f. 1/(1 + 4 * log(1-x)).

Original entry on oeis.org

1, 4, 36, 488, 8824, 199456, 5410208, 171209664, 6192052800, 251937937920, 11389639660032, 566394573855744, 30726758349800448, 1805828538127687680, 114293350061315678208, 7750480651439579529216, 560615413313367534698496, 43085423893717998388740096
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Column k=4 of A320079.
Cf. A094417, A354147, A354241.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+4*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=4*sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n, 4^k*k!*abs(stirling(n, k, 1)));

Formula

E.g.f.: 1/(1 + 4 * log(1-x)).
a(0) = 1; a(n) = 4 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).
a(n) = Sum_{k=0..n} 4^k * k! * |Stirling1(n, k)|.
a(n) ~ n! * exp(n/4) / (4 * (exp(1/4) - 1)^(n+1)). - Vaclav Kotesovec, Jun 04 2022
Showing 1-5 of 5 results.

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