A361840
Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 - x)^k)^(1/3).
Original entry on oeis.org
1, 1, 3, 1, 3, 18, 1, 3, 15, 126, 1, 3, 12, 90, 945, 1, 3, 9, 57, 585, 7371, 1, 3, 6, 27, 297, 3969, 58968, 1, 3, 3, 0, 78, 1629, 27657, 480168, 1, 3, 0, -24, -75, 207, 9216, 196290, 3961386, 1, 3, -3, -45, -165, -438, 459, 53217, 1411965, 33011550
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
3, 3, 3, 3, 3, 3, ...
18, 15, 12, 9, 6, 3, ...
126, 90, 57, 27, 0, -24, ...
945, 585, 297, 78, -75, -165, ...
7371, 3969, 1629, 207, -438, -444, ...
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T(n, k) = (-1)^n*sum(j=0, n, 9^j*binomial(-1/3, j)*binomial(k*j, n-j));
A361841
Expansion of 1/(1 - 9*x*(1+x)^2)^(1/3).
Original entry on oeis.org
1, 3, 24, 201, 1809, 16893, 161676, 1574289, 15527052, 154662930, 1552725504, 15688410264, 159355067283, 1625899880673, 16652520666414, 171119405299005, 1763475423260049, 18219685282559559, 188664151412242368, 1957539823296458841, 20347733657193596127
Offset: 0
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A361841 := n -> (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3, -n*2/3], [1/2 - n, 2/3 - n], -3/4):
seq(simplify(A361841(n)), n = 0..20); # Peter Luschny, Mar 27 2023
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my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1+x)^2)^(1/3))
A361842
Expansion of 1/(1 - 9*x*(1+x)^3)^(1/3).
Original entry on oeis.org
1, 3, 27, 243, 2352, 23607, 242757, 2539431, 26904492, 287858421, 3104029755, 33684914907, 367483636746, 4026930734223, 44295829667055, 488855016668727, 5410588668898995, 60035381850523284, 667643481187840206, 7439651232903588528, 83050643822779921347
Offset: 0
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a[n_]:=(-9)^n*Binomial[-1/3, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 2/3-n, 2/3-n}, -2^8/3^5]; Array[a,21,0] (* Stefano Spezia, Jul 11 2024 *)
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my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1+x)^3)^(1/3))
A361846
a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n*k,n-k).
Original entry on oeis.org
1, 3, 24, 243, 2973, 41676, 652662, 11228556, 209674050, 4211011422, 90309000630, 2056139084544, 49460437075896, 1251936022103679, 33228751234896060, 922028391785300940, 26676362307801924057, 802875670635086298600
Offset: 0
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a(n) = sum(k=0, n, (-9)^k*binomial(-1/3, k)*binomial(n*k, n-k));
Showing 1-4 of 4 results.