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.

A362354 a(n) = 3*(n+3)^(n-1).

Original entry on oeis.org

1, 3, 15, 108, 1029, 12288, 177147, 3000000, 58461513, 1289945088, 31813498119, 867763964928, 25949267578125, 844424930131968, 29713734098717811, 1124440102746243072, 45543381089624394897, 1966080000000000000000, 90125827485245075684223, 4372496892684322588065792
Offset: 0

Views

Author

Wolfdieter Lang, Apr 24 2023

Keywords

Comments

This gives the third exponential (also called binomial) convolution of {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. (LambertW(-x),(-x)) (LambertW is the principal branch of the Lambert W-function).
This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 3.

Links

Crossrefs

Column k=3 of A232006 (without leading zeros).
Cf. A137452.

Formula

a(n) = Sum_{k=0..n} |A137452(n, k)|*3^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*3^k, with the n = 0 term equal to 1 (not 0).
E.g.f.: (LambertW(-x)/(-x))^3.
From Seiichi Manyama, Jun 19 2024: (Start)
E.g.f. A(x) satisfies:
(1) A(x) = exp(3*x*A(x)^(1/3)).
(2) A(x) = 1/A(-x*A(x)^(2/3)). (End)

A362355 a(n) = 4*(n+4)^(n-1).

Original entry on oeis.org

1, 4, 24, 196, 2048, 26244, 400000, 7086244, 143327232, 3262922884, 82644187136, 2306601562500, 70368744177664, 2330488948919044, 83291859462684672, 3196026743131536484, 131072000000000000000, 5722274760967941313284, 264999811677837732610048
Offset: 0

Views

Author

Wolfdieter Lang, Apr 24 2023

Keywords

Comments

This gives the fourth exponential (also called binomial) convolution of {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. (LambertW(-x),(-x)) (LambertW is the principal branch of the Lambert W-function).
This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 4.

Links

Crossrefs

Column k = 4 of A232006 (without leading zeros).
Cf. A000272, A137452.

Programs

  • Mathematica
    Table[4(n+4)^(n-1),{n,0,20}] (* Harvey P. Dale, Jun 05 2024 *)

Formula

a(n) = Sum_{k=0..n} |A137452(n, k)|*4^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*4^k, with the n = 0 term equal to 1 (not 0).
E.g.f.: (LambertW(-x)/(-x))^4.
From Seiichi Manyama, Jun 19 2024: (Start)
E.g.f. A(x) satisfies:
(1) A(x) = exp(4*x*A(x)^(1/4)).
(2) A(x) = 1/A(-x*A(x)^(1/2)). (End)

A362356 a(n) = 5*(n + 5)^(n-1).

Original entry on oeis.org

1, 5, 35, 320, 3645, 50000, 805255, 14929920, 313742585, 7378945280, 192216796875, 5497558138880, 171359481538165, 5784156907130880, 210264917311285295, 8192000000000000000, 340611592914758411505, 15056807481695325716480, 705250197803314844630515
Offset: 0

Views

Author

Wolfdieter Lang, Apr 24 2023

Keywords

Comments

This gives the fifth exponential (also called binomial) convolution of {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. (LambertW(-x),(-x)) (LambertW is the principal branch of the Lambert W-function).
This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 5.

Links

Crossrefs

Column k=5 of A232006 (without leading zeros).
Cf. A000272, A137452.

Formula

a(n) = Sum_{k=0..n} |A137452(n, k)|*5^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*5^k, with the n = 0 term equal to 1 (not 0).
E.g.f.: (LambertW(-x)/(-x))^5.
From Seiichi Manyama, Jun 19 2024: (Start)
E.g.f. A(x) satisfies:
(1) A(x) = exp(5*x*A(x)^(1/5)).
(2) A(x) = 1/A(-x*A(x)^(2/5)). (End)

A384718 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A052750.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 12, 49, 0, 1, 4, 21, 128, 729, 0, 1, 5, 32, 243, 2000, 14641, 0, 1, 6, 45, 400, 3993, 41472, 371293, 0, 1, 7, 60, 605, 6912, 85683, 1075648, 11390625, 0, 1, 8, 77, 864, 10985, 153664, 2278125, 33554432, 410338673, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2025

Keywords

Examples

			Square array begins:
  1,     1,     1,     1,      1,      1, ...
  0,     1,     2,     3,      4,      5, ...
  0,     5,    12,    21,     32,     45, ...
  0,    49,   128,   243,    400,    605, ...
  0,   729,  2000,  3993,   6912,  10985, ...
  0, 14641, 41472, 85683, 153664, 253125, ...

Crossrefs

Columns k=0..2 give A000007, A052750, A097629(n+1).
Cf. A058127, A232006, A384692.

Programs

  • PARI
    a(n, k) = if(n==0, 1, k*(2*n+k)^(n-1));

Formula

A(n,k) = k * (2*n+k)^(n-1) for n > 0.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.