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.

A113551 a(n) = product of next n even numbers beginning with n if n is even, otherwise product of next n odd numbers beginning with n.

Original entry on oeis.org

1, 8, 105, 1920, 45045, 1290240, 43648605, 1703116800, 75293843625, 3719607091200, 203067496256625, 12140797545676800, 788917222956988125, 55362036808286208000, 4172583192219510193125, 336158287499913854976000
Offset: 1

Views

Author

Amarnath Murthy, Nov 03 2005

Keywords

Examples

			a(3) = 3*5*7 = 105, a(4) = 4*6*8*10 = 1920.

Links

Crossrefs

Cf. A113549, A113550, A303489.

Programs

  • Maple
    seq(mul((2*k+n), k=1..n)/3, n=1..16); # Zerinvary Lajos, Jan 29 2008
  • Mathematica
    Do[Print[Product[n + 2i, {i, 0, n - 1}]], {n, 1, 20}] (* Tracy Poff (tracy.poff(AT)gmail.com), Dec 31 2005 *)
Table[Times@@Range[n,3n-2,2],{n,20}] (* or *) Table[(2^n Gamma[(3n)/2])/Gamma[n/2],{n,20}] (* Harvey P. Dale, Nov 28 2022 *)
  • PARI
    for(n=1,25, print1(prod(k=0,n-1, n+2*k), ", ")) \\ G. C. Greubel, Sep 30 2017

Formula

a(n) = 2^n*Gamma(3*n/2)/Gamma(n/2). Column 2 of A303489. - Peter Bala, Jan 12 2020
a(n) ~ 3^((3*n-1)/2) * n^n / exp(n). - Vaclav Kotesovec, Aug 29 2024

Extensions

More terms from Tracy Poff (tracy.poff(AT)gmail.com), Dec 31 2005

A303486 a(n) = n! * [x^n] 1/(1 - 3*x)^(n/3).

Original entry on oeis.org

1, 1, 10, 162, 3640, 104720, 3674160, 152152000, 7264216960, 392841187200, 23734494784000, 1584471003315200, 115825295634048000, 9201578813819392000, 789383453851632640000, 72728093032166347776000, 7162140885524461957120000, 750766815289210771251200000
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

Examples

			a(1) = 1;
a(2) = 2*5 = 10;
a(3) = 3*6*9 = 162;
a(4) = 4*7*10*13 = 3640;
a(5) = 5*8*11*14*17 = 104720, etc.

Links

Crossrefs

Column k=3 of A303489.
Cf. A000407, A007559, A008544, A032031, A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609, A113551, A303487, A303488.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 - 3 x)^(n/3), {x, 0, n}], {n, 0, 17}]
Table[Product[3 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[3^n Pochhammer[n/3, n], {n, 0, 17}]

Formula

a(n) = Product_{k=0..n-1} (3*k + n).
a(n) = 3^n*Gamma(4*n/3)/Gamma(n/3).
a(n) ~ 2^(8*n/3-1)*n^n/exp(n).

A303487 a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).

Original entry on oeis.org

1, 1, 12, 231, 6144, 208845, 8648640, 422463195, 23781703680, 1515973484025, 107941254220800, 8491022274509775, 731304510986649600, 68444451854354701125, 6916953288171902976000, 750681472158682148959875, 87076954662428278259712000, 10751175443940144673035200625
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

Examples

			a(1) = 1;
a(2) = 2*6 = 12;
a(3) = 3*7*11 = 231;
a(4) = 4*8*12*16 = 6144;
a(5) = 5*9*13*17*21 = 208845, etc.

Links

Crossrefs

Column k=4 of A303489.
Cf. A000407, A001813, A007696, A008545, A034176, A034177, A047053, A051617, A051618, A051619, A051620, A051621, A051622, A113551, A303486, A303488.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 - 4 x)^(n/4), {x, 0, n}], {n, 0, 17}]
Table[Product[4 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[4^n Pochhammer[n/4, n], {n, 0, 17}]

Formula

a(n) = Product_{k=0..n-1} (4*k + n).
a(n) = 4^n*Gamma(5*n/4)/Gamma(n/4).
a(n) ~ 5^(5*n/4-1/2)*n^n/exp(n).

A303488 a(n) = n! * [x^n] 1/(1 - 5*x)^(n/5).

Original entry on oeis.org

1, 1, 14, 312, 9576, 375000, 17873856, 1004306688, 65006637696, 4763494479744, 389812500000000, 35237024762075136, 3487065897634615296, 374960171943074285568, 43532820293400237735936, 5427359437500000000000000, 723181462895975365595529216, 102563963819340862347122245632
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

Examples

			a(1) = 1;
a(2) = 2*7 = 14;
a(3) = 3*8*13 = 312;
a(4) = 4*9*14*19 = 9576;
a(5) = 5*10*15*20*25 = 375000, etc.

Links

Crossrefs

Column k=5 of A303489.
Cf. A008546, A008548, A034300, A034301, A034323, A034325, A047055, A047056, A051687, A051688, A051689, A051690, A051691, A052562, A113551, A303486, A303487.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 - 5 x)^(n/5), {x, 0, n}], {n, 0, 17}]
Table[Product[5 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[5^n Pochhammer[n/5, n], {n, 0, 17}]

Formula

a(n) = Product_{k=0..n-1} (5*k + n).
a(n) = 5^n*Gamma(6*n/5)/Gamma(n/5).
a(n) ~ 6^(6*n/5-1/2)*n^n/exp(n).

A131182 Table T(n,k) = n!*k^n, read by upwards antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 8, 3, 1, 0, 24, 48, 18, 4, 1, 0, 120, 384, 162, 32, 5, 1, 0, 720, 3840, 1944, 384, 50, 6, 1, 0, 5040, 46080, 29160, 6144, 750, 72, 7, 1, 0, 40320, 645120, 524880, 122880, 15000, 1296, 98, 8, 1, 0, 362880, 10321920, 11022480, 2949120, 375000, 31104, 2058, 128, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Sep 25 2007

Keywords

Comments

For k>0, T(n,k) is the n-th moment of the exponential distribution with mean = k. - Geoffrey Critzer, Jan 06 2019
T(n,k) is the minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}. For the maximum value, see A331988. - Chai Wah Wu, Sep 01 2022

Examples

			The (inverted) table begins:
k=0: 1, 0,   0,    0,      0,       0, ... (A000007)
k=1: 1, 1,   2,    6,     24,     120, ... (A000142)
k=2: 1, 2,   8,   48,    384,    3840, ... (A000165)
k=3: 1, 3,  18,  162,   1944,   29160, ... (A032031)
k=4: 1, 4,  32,  384,   6144,  122880, ... (A047053)
k=5: 1, 5,  50,  750,  15000,  375000, ... (A052562)
k=6: 1, 6,  72, 1296,  31104,  933120, ... (A047058)
k=7: 1, 7,  98, 2058,  57624, 2016840, ... (A051188)
k=8: 1, 8, 128, 3072,  98304, 3932160, ... (A051189)
k=9: 1, 9, 162, 4374, 157464, 7085880, ... (A051232)
Main diagonal is 1, 1, 8, 162, 6144, 375000, ... (A061711).

Links

Crossrefs

Columns k=0-9 give: A000007, A000142, A000165, A032031, A047053, A052562, A047058, A051188, A051189, A051232.
Main diagonal gives A061711.
Cf. A292783, A303489, A331988.

Programs

  • Maple
    T:= (n,k)-> n!*k^n:
seq(seq(T(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jan 06 2019
  • Python
    from math import factorial
def A131182_T(n, k): # compute T(n, k)
    return factorial(n)*k**n # Chai Wah Wu, Sep 01 2022

Formula

From Ilya Gutkovskiy, Aug 11 2017: (Start)
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - 2*k*x/(1 - 2*k*x/(1 - 3*k*x/(1 - 3*k*x/(1 - ...))))))), a continued fraction.
E.g.f. of column k: 1/(1 - k*x). (End)

A384216 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] (1 + k*x)^(n/k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -2, -3, 24, 1, 1, -4, 0, 0, 120, 1, 1, -6, 15, 40, 45, 720, 1, 1, -8, 42, 0, -280, 0, 5040, 1, 1, -10, 81, -264, -1155, 0, -1575, 40320, 1, 1, -12, 132, -896, 0, 20160, 24640, 0, 362880, 1, 1, -14, 195, -2040, 8645, 57456, -208845, -291200, 99225, 3628800
Offset: 0

Views

Author

Seiichi Manyama, May 22 2025

Keywords

Examples

			Square array begins:
    1,  1,    1,     1,     1,    1,       1, ...
    1,  1,    1,     1,     1,    1,       1, ...
    2,  0,   -2,    -4,    -6,   -8,     -10, ...
    6, -3,    0,    15,    42,   81,     132, ...
   24,  0,   40,     0,  -264, -896,   -2040, ...
  120, 45, -280, -1155,     0, 8645,   33120, ...
  720,  0,    0, 20160, 57456,    0, -459360, ...

Links

Crossrefs

Columns k=3..5 give A282627(n+1)/2, A383996, A383997.
Cf. A303489.

Programs

  • PARI
    a(n, k) = prod(j=0, n-1, n-k*j);

Formula

A(n,k) = Product_{j=0..n-1} (n-k*j).
A(n,k) = k^n * FallingFactorial(n/k,n).
A(k*n,k) = 0 for n > 0 and k > 1.
Showing 1-6 of 6 results.

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

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