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 1-3 of 3 results.

A061711 a(n) = n^n * n!.

Original entry on oeis.org

1, 1, 8, 162, 6144, 375000, 33592320, 4150656720, 676457349120, 140587147048320, 36288000000000000, 11388728893445164800, 4270826380475341209600, 1886009588552176549862400, 968725766854884321342259200, 572622616354851562500000000000
Offset: 0

Views

Author

Lorenzo Fortunato (fortunat(AT)pd.infn.it), Jun 19 2001

Keywords

Comments

a(n) is the product of first n terms of an arithmetic progression with first term n and common difference n. E.g. a(3) = 3*6*9 = 162. - Amarnath Murthy, Sep 20 2003
Product of the entries in the last column of an n X n square array whose elements are the numbers 1..n^2 listed in increasing order by rows. - Wesley Ivan Hurt, Mar 31 2025

Examples

			a(1) = 1^1 * 1! = 1;
a(2) = 2^2 * 2! = 8;
a(3) = 3^3 * 3! = 162.

Links

Crossrefs

Cf. A036679, A053042, A055775. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009
Main diagonal of A131182.
Cf. A336765.

Programs

  • Magma
    [Factorial(n)*n^n: n in [0..30]]; // G. C. Greubel, Nov 29 2022
  • Mathematica
    Table[If[n == 0, 1, n^n] * n!, {n, 0, 20}] (* Vaclav Kotesovec, Mar 08 2018 *)
  • PARI
    a(n) = n!*n^n; \\ Harry J. Smith, Jul 26 2009
  • Python
    from math import factorial
def A061711(n): return factorial(n)*n**n # Chai Wah Wu, Sep 03 2022

Formula

E.g.f.: sinh(n*x)^n. - Vaclav Kotesovec, Nov 05 2014
a(n) = [x^n] 1/(1 - n*x/(1 - n*x/(1 - 2*n*x/(1 - 2*n*x/(1 - 3*n*x/(1 - 3*n*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Sep 20 2017
Sum_{n>=1} 1/a(n) = A336765. - Amiram Eldar, Nov 20 2020
a(n) ~ exp(-n)*n^(2*n)*sqrt(2*n*Pi). - Peter Luschny, Jan 10 2022

A303489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 8, 60, 1, 1, 10, 105, 840, 1, 1, 12, 162, 1920, 15120, 1, 1, 14, 231, 3640, 45045, 332640, 1, 1, 16, 312, 6144, 104720, 1290240, 8648640, 1, 1, 18, 405, 9576, 208845, 3674160, 43648605, 259459200, 1, 1, 20, 510, 14080, 375000, 8648640, 152152000, 1703116800, 8821612800
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

Examples

			Square array begins:
      1,      1,       1,       1,       1,       1,  ...
      1,      1,       1,       1,       1,       1,  ...
      6,      8,      10,      12,      14,      16,  ...
     60,    105,     162,     231,     312,     405,  ...
    840,   1920,    3640,    6144,    9576,   14080,  ...
  15120,  45045,  104720,  208845,  375000,  623645,  ...
=========================================================
A(1,1) = 1;
A(2,1) = 2*3 = 6;
A(3,1) = 3*4*5 = 60;
A(4,1) = 4*5*6*7 = 840;
A(5,1) = 5*6*7*8*9 = 15120, etc.
...
A(1,2) = 1;
A(2,2) = 2*4 = 8;
A(3,2) = 3*5*7 = 105;
A(4,2) = 4*6*8*10 = 1920;
A(5,2) = 5*7*9*11*13 = 45045, etc.
...
A(1,3) = 1;
A(2,3) = 2*5 = 10;
A(3,3) = 3*6*9 = 162;
A(4,3) = 4*7*10*13 = 3640;
A(5,3) = 5*8*11*14*17 = 104720, etc.
...

Links

Crossrefs

Columns k=1..5 give A000407, A113551, A303486, A303487, A303488.
Main diagonal gives A061711.
Cf. A008279, A131182, A256268, A265609.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[1/(1 - k x)^(n/k), {x, 0, n}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, Product[k i + n, {i, 0, n - 1}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, k^n Pochhammer[n/k, n]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten

Formula

A(n,k) = Product_{j=0..n-1} (k*j + n).

A292783 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/sqrt(1 - 2*k*x).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 12, 15, 0, 1, 4, 27, 120, 105, 0, 1, 5, 48, 405, 1680, 945, 0, 1, 6, 75, 960, 8505, 30240, 10395, 0, 1, 7, 108, 1875, 26880, 229635, 665280, 135135, 0, 1, 8, 147, 3240, 65625, 967680, 7577955, 17297280, 2027025, 0, 1, 9, 192, 5145, 136080, 2953125, 42577920, 295540245, 518918400, 34459425, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2017

Keywords

Examples

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + 3*k^2*x^2/2! + 15*k^3*x^3/3! + 105*k^4*x^4/4! + 945*k^5*x^5/5! + 10395*k^6*x^6/6! +
Square array begins:
1,    1,      1,       1,       1,        1,  ...
0,    1,      2,       3,       4,        5,  ...
0,    3,     12,      27,      48,       75,  ...
0,   15,    120,     405,     960,     1875,  ...
0,  105,   1680,    8505,   26880,    65625,  ...
0,  945,  30240,  229635,  967680,  2953125,  ...

Crossrefs

Columns k=0..4 give A000007, A001147, A001813, A011781, A144828.
Rows n=0.2 give A000012, A001477, A033428.
Main diagonal gives A292784.
Cf. A131182.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[1/Sqrt[1 - 2 k x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-i k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

O.g.f. of column k: 1/(1 - k*x/(1 - 2*k*x/(1 - 3*k*x/(1 - 4*k*x/(1 - 5*k*x/(1 - ...)))))), a continued fraction.
E.g.f. of column k: 1/sqrt(1 - 2*k*x).
A(n,k) = k^n*A001147(n).
Showing 1-3 of 3 results.

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