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 1-8 of 8 results.

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).

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).

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).

A384166 a(n) = Product_{k=0..n-1} (3*n+4*k).

Original entry on oeis.org

1, 3, 60, 1989, 92160, 5486535, 399072960, 34298042625, 3400783626240, 382128386114475, 47986411423104000, 6659996213472126525, 1012334387351519232000, 167253493686752981883375, 29842935065036371998720000, 5719198821953333723419037625, 1171620424982972483984424960000
Offset: 0

Views

Author

Seiichi Manyama, May 21 2025

Keywords

Links

Crossrefs

Cf. A384164, A384165.
Cf. A303487.

Programs

  • Magma
    [1] cat  [&*[(3*n + 4*k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025
  • Mathematica
    a[n_]:=Product[(3*n+4*k),{k,0,n-1}];Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)
  • PARI
    a(n) = prod(k=0, n-1, 3*n+4*k);
  • Python
    from math import prod
def A384166(n): return prod(3*n+i for i in range(0,n<<2,4)) # Chai Wah Wu, May 21 2025
  • Sage
    def a(n): return 4^n*rising_factorial(3*n/4, n)

Formula

a(n) = 4^n * RisingFactorial(3*n/4,n).
a(n) = n! * [x^n] 1/(1 - 4*x)^(3*n/4).
a(n) = (3/7) * 4^n * n! * binomial(7*n/4,n) for n > 0.

A383996 a(n) = Product_{k=0..n-1} (n-4*k).

Original entry on oeis.org

1, 1, -4, 15, 0, -1155, 20160, -208845, 0, 68139225, -2075673600, 34976316375, 0, -25949801752875, 1126343522304000, -26264240610733125, 0, 34770736214117528625, -1958486116582195200000, 58318039100493206409375, 0, -120842042784862988395681875, 8366746697372733839769600000
Offset: 0

Views

Author

Seiichi Manyama, May 22 2025

Keywords

Crossrefs

Column k=4 of A384216.
Cf. A303487.

Programs

  • PARI
    a(n) = prod(k=0, n-1, n-4*k);
  • Sage
    def a(n): return 4^n*falling_factorial(n/4, n)

Formula

a(n) = 4^n * FallingFactorial(n/4,n).
a(n) = n! * [x^n] (1 + 4*x)^(n/4).
a(4*n) = 0 for n > 0.

A384169 a(n) = 4^n * n! * binomial(5*n/4,n) * Sum_{k=1..n} 1/(n+4*k).

Original entry on oeis.org

1, 16, 347, 9856, 349269, 14885760, 742589175, 42479124480, 2742327328905, 197267905658880, 15649214440432275, 1357388618032742400, 127808331929417605725, 12983375200126773657600, 1415428114244995252270575, 164837363498660501913600000, 20423530465926352502482292625
Offset: 1

Views

Author

Seiichi Manyama, May 21 2025

Keywords

Crossrefs

Cf. A098118, A384167, A384168.
Cf. A303487.

Programs

  • PARI
    a(n) = sum(k=0, n, k*(n+4)^(k-1)*4^(n-k)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} k * (n+4)^(k-1) * 4^(n-k) * |Stirling1(n,k)|.
a(n) = n! * [x^n] ( -log(1 - 4*x)/(4 * (1 - 4*x)^(n/4+1)) ).

A384258 a(n) = Product_{k=0..n-1} (n+4*k+1).

Original entry on oeis.org

1, 2, 21, 384, 9945, 332640, 13627845, 660602880, 36974963025, 2346549004800, 166490632833525, 13059009124761600, 1122040194333683625, 104802322548059136000, 10572978481108199281125, 1145749403453003661312000, 132730561036298082383150625, 16369108295524571830763520000
Offset: 0

Views

Author

Seiichi Manyama, May 23 2025

Keywords

Crossrefs

Cf. A303487, A384259.

Programs

  • PARI
    a(n) = prod(k=0, n-1, n+4*k+1);
  • Sage
    def a(n): return 4^n*rising_factorial((n+1)/4, n)

Formula

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

A384259 a(n) = Product_{k=0..n-1} (n+4*k+3).

Original entry on oeis.org

1, 4, 45, 840, 21945, 737280, 30282525, 1470268800, 82380323025, 5231974809600, 371413503586125, 29144138639616000, 2504851570980383625, 234017443515727872000, 23613335889752371888125, 2559272716623604101120000, 296519181502679448839150625, 36572320958219876869079040000
Offset: 0

Views

Author

Seiichi Manyama, May 23 2025

Keywords

Crossrefs

Cf. A303487, A384258.

Programs

  • PARI
    a(n) = prod(k=0, n-1, n+4*k+3);
  • Sage
    def a(n): return 4^n*rising_factorial((n+3)/4, n)

Formula

a(n) = 4^n * RisingFactorial((n+3)/4,n).
a(n) = n! * [x^n] 1/(1 - 4*x)^((n+3)/4).
D-finite with recurrence a(n) -5*(5*n-9)*(5*n-13)*(5*n-1)*(5*n-17)*a(n-4)=0. - R. J. Mathar, May 26 2025
Showing 1-8 of 8 results.

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

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