A360291
a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2*k,k) * binomial(2*n-6*k,n-3*k).
Original entry on oeis.org
1, 2, 6, 20, 72, 264, 984, 3714, 14148, 54284, 209482, 812196, 3161340, 12345658, 48348522, 189807336, 746740510, 2943359208, 11620961412, 45950375602, 181936110006, 721233025332, 2862271873966, 11370584735100, 45212101270728, 179926167512914
Offset: 0
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a(n) = sum(k=0, n\3, binomial(n-1-2*k, k)*binomial(2*n-6*k, n-3*k));
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my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^3)))
A360292
a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3*k,k) * binomial(2*n-8*k,n-4*k).
Original entry on oeis.org
1, 2, 6, 20, 70, 254, 936, 3492, 13150, 49882, 190318, 729576, 2807816, 10841962, 41983588, 162973568, 633994982, 2471010742, 9646981054, 37718873700, 147676286078, 578883674722, 2271704404900, 8923807316892, 35087269756344, 138075819924306
Offset: 0
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a(n) = sum(k=0, n\4, binomial(n-1-3*k, k)*binomial(2*n-8*k, n-4*k));
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my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^4)))
A383573
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(2*(n-2*k),n-2*k).
Original entry on oeis.org
1, 2, 7, 24, 89, 338, 1311, 5152, 20449, 81778, 328999, 1330008, 5398265, 21984610, 89791103, 367643776, 1508560257, 6201927074, 25540266503, 105336838616, 435035342553, 1798875915826, 7446653956895, 30857577536800, 127987031688161, 531301328367762, 2207281722474919
Offset: 0
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[&+[Binomial(n-k, k) * Binomial(2*(n-2*k), n-2*k): k in [0..Floor(n div 2)]]: n in [0..35]]; // Vincenzo Librandi, May 03 2025
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Table[Sum[Binomial[n-k,k]* Binomial[2*(n-2*k),n-2*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 03 2025 *)
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a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(2*(n-2*k), n-2*k));
Showing 1-3 of 3 results.