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.

A320083 Expansion of e.g.f. Sum_{k>=0} log(1 + k*x)^k.

Original entry on oeis.org

1, 1, 7, 116, 3574, 177094, 12873962, 1290494904, 170592253320, 28753159552272, 6018433850602848, 1531605185388897552, 465706857941949607008, 166746568516127626614288, 69440517484503283491716400, 33278913978673363553703249408, 18185279212166784139689388753536, 11239676837467731657648932618952576
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 05 2018

Keywords

Links

Crossrefs

Cf. A048594, A048994, A122399, A317172, A320096.

Programs

  • Maple
    1,seq(n!*coeff(series(add( log(1 + k*x)^k,k=1..100), x=0, 18), x, n), n=1..17); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 17; CoefficientList[Series[1 + Sum[Log[1 + k x]^k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k! k^n, {k, n}], {n, 17}]]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*k!*k^n.
a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.298212940253960977992575968955431001807757948758929... and c = 3.40415549717199390989204785905061856492539214306... - Vaclav Kotesovec, Oct 05 2018

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

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

A352074 a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).

Original entry on oeis.org

1, 1, 4, 42, 904, 34070, 2019888, 174588120, 20804747136, 3276218158560, 659664288364800, 165425062846302336, 50574549124825998336, 18520126461205806360144, 8003819275469728355033088, 4031020344281171589447408000, 2340375822778055527109749211136
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 02 2022

Keywords

Links

Crossrefs

Cf. A007840, A092985, A094420, A227917, A317171, A317172, A331690, A352069, A352071.
Cf. A081048.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! (-n)^(n - k), {k, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[1/(1 + Log[1 - n x]/n), {x, 0, n}], {n, 1, 16}]]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 1)*k!*(-n)^(n-k)); \\ Michel Marcus, Mar 02 2022

Formula

a(n) = n! * [x^n] 1 / (1 + log(1 - n*x) / n) for n > 0.
a(n) ~ n! * n^(n-2) * (1 + 2*log(n)/n). - Vaclav Kotesovec, Mar 03 2022

A335529 a(n) = n! * [x^n] (1 - (n-1)*log(1 + x))/(1 - n*log(1 + x)).

Original entry on oeis.org

1, 1, 3, 38, 1042, 49774, 3661128, 383653080, 54275300112, 9964363066848, 2303245150868640, 654457584668128416, 224205104879416320768, 91129285853151907958544, 43356207229026959513863680, 23868203329368882698589532800, 15053662436260897659550535387136
Offset: 0

Views

Author

Seiichi Manyama, Jun 12 2020

Keywords

Crossrefs

Main diagonal of A334369.
Cf. A317172, A321189.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * n^(k - 1) * StirlingS1[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 12 2020 *)
  • PARI
    {a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 1)))}

Formula

a(n) = A317172(n)/n = Sum_{k=0..n} k!*n^(k-1)*Stirling1(n,k) for n > 1.
a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jun 12 2020

A354752 a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * n^(n-k).

Original entry on oeis.org

1, 1, 0, 6, -152, 6670, -451152, 43685208, -5741360256, 984176280288, -213379094227200, 57100689621382176, -18489130293293779968, 7125765731670143814672, -3223822934974620319272960, 1692009521117003600170128000, -1019755541584493644326799048704
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2022

Keywords

Links

Crossrefs

Cf. A006252, A094420, A317171, A317172, A331690, A349731, A352074, A354237, A354750, A354751.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! n^(n - k), {k, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[1/(1 - Log[1 + n x]/n), {x, 0, n}], {n, 1, 16}]]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 1) * k! * n^(n-k)); \\ Michel Marcus, Jun 06 2022

Formula

a(n) = n! * [x^n] 1 / (1 - log(1 + n*x) / n) for n > 0.
a(n) ~ (-1)^(n+1) * n! * n^(n-2). - Vaclav Kotesovec, Jun 06 2022
Showing 1-6 of 6 results.

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