A269661
a(n) = Product_{i=1..n} (5^i - 4^i).
Original entry on oeis.org
1, 9, 549, 202581, 425622681, 4907003889249, 302963327126122509, 98490045052104040328301, 166544794872251942218390753281, 1451779137596368920662880897497387769, 64798450159010700654830227323217753649135349
Offset: 1
Cf. sequences of the form Product_{i=1..n}(j^i - 1):
A005329 (j=2),
A027871 (j=3),
A027637 (j=4),
A027872 (j=5),
A027873 (j=6),
A027875 (j=7),
A027876 (j=8),
A027877 (j=9),
A027878 (j=10),
A027879 (j=11),
A027880 (j=12).
Cf. sequences of the form Product_{i=1..n}(j^i - k^1), k>1:
A263394 (j=3, k=2),
A269576 (j=4, k=3).
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[&*[ 5^k-4^k: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Mar 03 2016
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Table[Product[5^i - 4^i, {i, n}], {n, 15}] (* Vincenzo Librandi, Mar 03 2016 *)
Table[5^(Binomial[n + 1, 2]) *QPochhammer[4/5, 4/5, n], {n, 1, 20}] (* G. C. Greubel, Mar 05 2016 *)
FoldList[Times,Table[5^n-4^n,{n,15}]] (* Harvey P. Dale, Aug 28 2018 *)
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a(n) = prod(k=1, n, 5^k-4^k); \\ Michel Marcus, Mar 05 2016
A320354
Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Product_{j=1..n} (k^j - 1).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 16, 21, 0, 1, 4, 45, 416, 315, 0, 1, 5, 96, 2835, 33280, 9765, 0, 1, 6, 175, 11904, 722925, 8053760, 615195, 0, 1, 7, 288, 37625, 7428096, 739552275, 5863137280, 78129765, 0, 1, 8, 441, 98496, 48724375, 23205371904, 3028466566125, 12816818094080, 19923090075, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 3, 16, 45, 96, 175, ...
0, 21, 416, 2835, 11904, 37625, ...
0, 315, 33280, 722925, 7428096, 48724375, ...
0, 9765, 8053760, 739552275, 23205371904, 378832015625, ...
Columns k=1..12 give
A000007,
A005329,
A027871,
A027637,
A027872,
A027873,
A027875,
A027876,
A027877,
A027878,
A027879,
A027880.
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Table[Function[k, Product[k^j - 1, {j, 1, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[k^(i (i + 1)/2) x^i/Product[(1 + k^j x), {j, 0, i}], {i, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten