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
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.
Cf.
A000407,
A007559,
A008544,
A032031,
A034000,
A034001,
A051604,
A051605,
A051606,
A051607,
A051608,
A051609,
A113551,
A303487,
A303488.
-
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}]
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
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.
Cf.
A000407,
A001813,
A007696,
A008545,
A034176,
A034177,
A047053,
A051617,
A051618,
A051619,
A051620,
A051621,
A051622,
A113551,
A303486,
A303488.
-
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}]
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
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.
...
-
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
A383997
a(n) = Product_{k=0..n-1} (n-5*k).
Original entry on oeis.org
1, 1, -6, 42, -264, 0, 57456, -1808352, 40715136, -643458816, 0, 583285038336, -32763345398784, 1237080874917888, -31193431756591104, 0, 64105508174249558016, -5177532237241354518528, 274167069135623993032704, -9487174826303791319678976, 0
Offset: 0
-
a(n) = prod(k=0, n-1, n-5*k);
-
def a(n): return 5^n*falling_factorial(n/5, n)
Showing 1-4 of 4 results.