A386649
Product of first n central trinomial coefficients (A002426) for n > 0 with a(0) = 1.
Original entry on oeis.org
1, 1, 3, 21, 399, 20349, 2869209, 1127599137, 1248252244659, 3918263795984601, 35080215765450132753, 899912775031092255512709, 66403663756769266442027284401, 14140062564030204365431731967633341, 8713488333644640745496899895218790824407
Offset: 0
The central trinomial coefficients A002426(n) = [x^n] (1 + x + x^2)^n for n >= 0 begin [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, ...], where a(n) equals the product of the terms A002426(0) through A002426(n).
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Table[Product[3^k * Hypergeometric2F1[1/2, -k, 1, 4/3], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 09 2025 *)
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{a(n) = prod(k=0,n, polcoef((1 + x + x^2)^k, k) )}
for(n=0,15,print1(a(n),", "))
A386650
Product of first n quadrinomial coefficients (A005725) for n > 0 with a(0) = 1.
Original entry on oeis.org
1, 1, 3, 30, 930, 93930, 31560480, 35600221440, 136099646565120, 1776236487321381120, 79580723341459838319360, 12296654209275691297430868480, 6578267322410960919238807125534720, 12223446894741861497849104893155093176320, 79112201841847644246811045518121813092796661760
Offset: 0
The quadrinomial coefficients A005725(n) = [x^n] (1 + x + x^2 + x^3)^n for n >= 0 begin [1, 1, 3, 10, 31, 101, 336, 1128, 3823, ...], where a(n) equals the product of the terms A005725(0) through A005725(n).
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Table[Product[HypergeometricPFQ[{(1-k)/2, -k, -k/2}, {1/2, 1}, -1], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 09 2025 *)
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{a(n) = prod(k=0, n, polcoef((1 + x + x^2 + x^3)^k, k) )}
for(n=0, 15, print1(a(n), ", "))
A342178
Product of first n central Delannoy numbers.
Original entry on oeis.org
1, 3, 39, 2457, 788697, 1327377051, 11931792311439, 580350446236081521, 154215943727867706493809, 225550533306461376412704772467, 1826384842574005591817185497927226551, 82272644789290466599017454496002856892236169
Offset: 0
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b:= proc(n) option remember; `if`(n<1, 1,
(3*(2*n-1)*b(n-1) -(n-1)*b(n-2))/n)
end:
a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
seq(a(n), n=0..15); # Alois P. Heinz, Mar 04 2021
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Table[Product[Hypergeometric2F1[-k, k+1, 1, -1], {k, 1, n}], {n, 0, 15}]
FoldList[Times, 1, Table[Hypergeometric2F1[-n, n + 1, 1, -1], {n, 1, 15}]]
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D(n) = sum(k=0, n, binomial(n, k)*binomial(n+k, k)); \\ A001850
a(n) = prod(k=0, n, D(k)); \\ Michel Marcus, Mar 04 2021
Showing 1-3 of 3 results.
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