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));
A383582
a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*(n-4*k),n-4*k).
Original entry on oeis.org
1, 2, 6, 20, 71, 256, 942, 3512, 13221, 50138, 191260, 733088, 2821037, 10892100, 42174848, 163706656, 636816019, 2481902842, 9689155902, 37882580356, 148313102097, 581365577564, 2281393560802, 8961689897248, 35235582858441, 138657185501870, 546064549476476
Offset: 0
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[&+[Binomial(n-3*k,k) * Binomial(2*(n-4*k),n-4*k): k in [0..n div 4]]: n in [0..45]]; // Vincenzo Librandi, May 02 2025
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Table[Sum[Binomial[n-3*k,k]* Binomial[2*(n-4*k),n-4*k],{k,0,Floor[n/4]}],{n,0,30}] (* Vincenzo Librandi, May 02 2025 *)
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a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*(n-4*k), n-4*k));
A383572
Expansion of 1/sqrt((1-x^4)^2 - 4*x^5).
Original entry on oeis.org
1, 0, 0, 0, 1, 2, 0, 0, 1, 6, 6, 0, 1, 12, 30, 20, 1, 20, 90, 140, 71, 30, 210, 560, 631, 294, 420, 1680, 3151, 2828, 1680, 4200, 11551, 16704, 13272, 12672, 34651, 72162, 86064, 69960, 102961, 252362, 423390, 446160, 429001, 805508, 1685970, 2393820, 2419561
Offset: 0
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a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(k, n-4*k));
A383566
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 (1,0),(0,1),(4,4).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 71, 56, 28, 8, 1, 1, 9, 36, 84, 128, 128, 84, 36, 9, 1, 1, 10, 45, 120, 213, 258, 213, 120, 45, 10, 1, 1, 11, 55, 165, 334, 474, 474, 334, 165, 55, 11, 1
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 3, 6, 10, 15, 21, 28, ...
1, 4, 10, 20, 35, 56, 84, ...
1, 5, 15, 35, 71, 128, 213, ...
1, 6, 21, 56, 128, 258, 474, ...
1, 7, 28, 84, 213, 474, 954, ...
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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-y-x^4*y^4), n), k);
Showing 1-4 of 4 results.
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