A376791
Expansion of 1/sqrt((1 - x^3)^2 - 4*x).
Original entry on oeis.org
1, 2, 6, 21, 76, 282, 1065, 4074, 15732, 61193, 239406, 941064, 3713701, 14703896, 58383138, 232383841, 926943678, 3704410890, 14828984641, 59450138412, 238659074286, 959247218253, 3859777477944, 15546444564846, 62675854384977, 252893414725842, 1021208266423260
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-x^3)^2-4*x))
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a(n) = sum(k=0, n\3, binomial(2*n-5*k, k)*binomial(2*n-6*k, n-3*k));
A383571
Expansion of 1/sqrt((1-x^3)^2 - 4*x^4).
Original entry on oeis.org
1, 0, 0, 1, 2, 0, 1, 6, 6, 1, 12, 30, 21, 20, 90, 141, 100, 210, 561, 672, 672, 1681, 3206, 3528, 5125, 11622, 17892, 21253, 38172, 74052, 102565, 141680, 268092, 454741, 622182, 979836, 1790361, 2784366, 3993132, 6741593, 11587758, 17380116, 26551097, 45489082, 74098518
Offset: 0
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a(n) = sum(k=0, n\3, binomial(n-2*k, k)*binomial(k, n-3*k));
A383568
Expansion of 1/sqrt((1-x^5)^2 - 4*x^2).
Original entry on oeis.org
1, 0, 2, 0, 6, 1, 20, 6, 70, 30, 253, 140, 936, 630, 3522, 2773, 13430, 12032, 51770, 51690, 201389, 220470, 789546, 935330, 3116416, 3951949, 12373910, 16645398, 49389050, 69938416, 198048409, 293296470, 797461358, 1228136090, 3222960100, 5136602753
Offset: 0
A383550
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (2,0),(0,2),(3,3).
Original entry on oeis.org
1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 2, 6, 2, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 3, 10, 6, 10, 3, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 4, 15, 12, 21, 12, 15, 4, 6, 0, 1
Offset: 0
Square array A(n,k) begins:
1, 0, 1, 0, 1, 0, 1, 0, 1, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 2, 0, 3, 0, 4, 0, 5, ...
0, 0, 0, 1, 0, 2, 0, 3, 0, ...
1, 0, 3, 0, 6, 0, 10, 0, 15, ...
0, 0, 0, 2, 0, 6, 0, 12, 0, ...
1, 0, 4, 0, 10, 0, 21, 0, 38, ...
0, 0, 0, 3, 0, 12, 0, 30, 0, ...
1, 0, 5, 0, 15, 0, 38, 0, 82, ...
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a(n, k) = my(x='x+O('x^(n+1)), y='y+O('y^(k+1))); polcoef(polcoef(1/(1-x^2-y^2-x^3*y^3), n), k);
A383569
Expansion of 1/sqrt((1-x^7)^2 - 4*x^2).
Original entry on oeis.org
1, 0, 2, 0, 6, 0, 20, 1, 70, 6, 252, 30, 924, 140, 3433, 630, 12882, 2772, 48710, 12012, 185316, 51481, 708582, 218810, 2720788, 923990, 10484684, 3881556, 40528441, 16236486, 157086660, 67675972, 610318610, 281236620, 2376289056, 1165715161, 9269869182
Offset: 0
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CoefficientList[Series[1/Sqrt[(1-x^7)^2-4x^2],{x,0,40}],x] (* Harvey P. Dale, Aug 09 2025 *)
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my(N=40, x='x+O('x^N)); Vec(1/sqrt((1-x^7)^2-4*x^2))
Showing 1-5 of 5 results.
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