A361812
Expansion of 1/sqrt(1 - 4*x*(1+x)^3).
Original entry on oeis.org
1, 2, 12, 62, 342, 1932, 11094, 64480, 378150, 2233304, 13263772, 79136844, 473969586, 2847911596, 17159547804, 103640073972, 627280131594, 3803643145596, 23102172930156, 140522319418164, 855880464524472, 5219168576004184, 31861229045809436
Offset: 0
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a[n_]:=Binomial[2*n, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 1/2-n, 2/3-n}, -2^6/3^3]; Array[a,23,0] (* Stefano Spezia, Jul 11 2024 *)
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my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^3))
A361814
Expansion of 1/sqrt(1 - 4*x*(1+x)^5).
Original entry on oeis.org
1, 2, 16, 100, 660, 4482, 30886, 215364, 1515000, 10730800, 76426846, 546792056, 3926775646, 28290272420, 204375145480, 1479963148220, 10739326203132, 78072933869364, 568503202324540, 4145718464390120, 30271771382355430, 221305746414518180
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^5))
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a(n)= sum(k=0, n, binomial(2*k,k) * binomial(5*k,n-k)) \\ Winston de Greef, Mar 25 2023
A361830
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2*j,j) * binomial(k*j,n-j).
Original entry on oeis.org
1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 10, 32, 70, 1, 2, 12, 46, 136, 252, 1, 2, 14, 62, 226, 592, 924, 1, 2, 16, 80, 342, 1136, 2624, 3432, 1, 2, 18, 100, 486, 1932, 5810, 11776, 12870, 1, 2, 20, 122, 660, 3030, 11094, 30080, 53344, 48620
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, ...
6, 8, 10, 12, 14, 16, ...
20, 32, 46, 62, 80, 100, ...
70, 136, 226, 342, 486, 660, ...
252, 592, 1136, 1932, 3030, 4482, ...
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T(n, k) = sum(j=0, n, binomial(2*j, j)*binomial(k*j, n-j));
A361817
Expansion of 1/sqrt(1 - 4*x*(1-x)^4).
Original entry on oeis.org
1, 2, -2, -16, -10, 118, 304, -500, -3754, -2488, 30866, 83716, -135568, -1080972, -792876, 9090484, 25788118, -39325156, -335074520, -271779024, 2820643842, 8348113120, -11788972644, -107836934448, -96107852032, 900943403012, 2778574561276, -3596374190416
Offset: 0
Showing 1-4 of 4 results.