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.

A382800 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (1 - log(1-x) * log(1-y))^3.

Original entry on oeis.org

1, 0, 0, 0, 3, 0, 0, 3, 3, 0, 0, 6, 27, 6, 0, 0, 18, 78, 78, 18, 0, 0, 72, 282, 588, 282, 72, 0, 0, 360, 1272, 2988, 2988, 1272, 360, 0, 0, 2160, 6936, 16344, 24612, 16344, 6936, 2160, 0, 0, 15120, 44496, 101448, 175632, 175632, 101448, 44496, 15120, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2025

Keywords

Examples

			Square array begins:
  1,  0,    0,     0,      0,       0, ...
  0,  3,    3,     6,     18,      72, ...
  0,  3,   27,    78,    282,    1272, ...
  0,  6,   78,   588,   2988,   16344, ...
  0, 18,  282,  2988,  24612,  175632, ...
  0, 72, 1272, 16344, 175632, 1669128, ...

Crossrefs

Main diagonal gives A382806.
Cf. A379821, A382799.
Cf. A382735, A382802.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*abs(stirling(n, j, 1)*stirling(k, j, 1)));

Formula

E.g.f.: 1 / (1 - log(1-x) * log(1-y))^3.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * |Stirling1(n,j)| * |Stirling1(k,j)|.

A382804 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling1(n,k)^2.

Original entry on oeis.org

1, 2, 14, 260, 9588, 581952, 52096512, 6423520896, 1041005447424, 214260350714496, 54547409318781312, 16820040059243046144, 6175245603727007034624, 2661063379044058584861696, 1329787781176741647226481664, 762665713456216694195942866944
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2025

Keywords

Crossrefs

Main diagonal of A382799.
Cf. A382792, A382806.
Cf. A382737.

Programs

  • PARI
    a(n) = sum(k=0, n, k!*(k+1)!*stirling(n, k, 1)^2);

Formula

a(n) == 0 (mod 2) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1-x) * log(1-y))^2.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1+x) * log(1+y))^2.
a(n) ~ sqrt(Pi) * n^(2*n + 3/2) / (exp(1) - 1)^(2*n+2). - Vaclav Kotesovec, Apr 05 2025

A382828 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling1(n+1,k+1)^2.

Original entry on oeis.org

1, 4, 55, 1623, 82116, 6302028, 680105112, 98011315608, 18163969766592, 4205977241171328, 1189459906531372224, 403300593144673493184, 161454763431242385682176, 75337361633768810384542464, 40524573487904551618353921024, 24890567631479746511661428751360
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2025

Keywords

Crossrefs

Main diagonal of A382825.
Cf. A382676, A382806.

Programs

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

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^3 ).
a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1+x) * (1+y) * (1 - log(1+x) * log(1+y))^3 ).

A382853 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (k! * Stirling1(n,k))^2.

Original entry on oeis.org

1, 1, 14, 588, 51064, 7542780, 1688795184, 532244030976, 224335607135616, 121793234373123840, 82750681453274478720, 68773648886955417943296, 68628724852793337500166144, 80970628401965472953705395200, 111490683570184861858636405923840, 177177650274516448010905794637332480
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2025

Keywords

Crossrefs

Cf. A382792, A382804, A382806.

Programs

  • Mathematica
    Table[Sum[Binomial[n+k-1,k] * k!^2 * StirlingS1[n,k]^2, {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 07 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 1))^2);

Formula

a(n) == 0 (mod n) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1-x) * log(1-y))^n.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1+x) * log(1+y))^n.
a(n) ~ c * (r*(1+r) + sqrt(r*(1+r)))^(2*n) * n^(2*n) / (exp(2*n) * r^n), where r = 0.71197519729041875298209529969157574831688314013967... is the root of the equation (1+r)*(r + LambertW(-1, -r*exp(-r)))^2 = r and c = 0.61294561390083215776201123658816241786650851195222... - Vaclav Kotesovec, Apr 07 2025
Showing 1-4 of 4 results.

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