A360049
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+2,3*k+2) * Catalan(k).
Original entry on oeis.org
1, 3, 6, 9, 9, 0, -26, -72, -117, -82, 204, 975, 2289, 3357, 1332, -9834, -37935, -82593, -108282, 2583, 487521, 1621071, 3261546, 3685230, -2318615, -24607854, -72887472, -134909701, -123941901, 200330184, 1258932996, 3377359872, 5706502677, 3797618237
Offset: 0
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Table[Sum[(-1)^k Binomial[n+2,3k+2]CatalanNumber[k],{k,0,Floor[n/3]}],{n,0,40}] (* Harvey P. Dale, Nov 21 2023 *)
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a(n) = sum(k=0, n\3, (-1)^k*binomial(n+2, 3*k+2)*binomial(2*k, k)/(k+1));
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my(N=40, x='x+O('x^N)); Vec(2/((1-x)*((1-x)^2+sqrt((1-x)^4+4*x^3*(1-x)))))
A360051
a(n) = Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+4,5*k+4) * Catalan(k).
Original entry on oeis.org
1, 5, 15, 35, 70, 125, 200, 275, 275, 0, -999, -3610, -9380, -20570, -39440, -65251, -85695, -56435, 141735, 781770, 2413128, 5999325, 12921350, 24387900, 39098925, 46638744, 11740695, -158571665, -674961760, -1956733020, -4724183860, -9957286550, -18316004575
Offset: 0
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a(n) = sum(k=0, n\5, (-1)^k*binomial(n+4, 5*k+4)*binomial(2*k, k)/(k+1));
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my(N=40, x='x+O('x^N)); Vec(2/((1-x)^2*((1-x)^3+sqrt((1-x)^6+4*x^5*(1-x)))))
A360048
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+1,2*k+1) * Catalan(k).
Original entry on oeis.org
1, 2, 2, 0, -3, -2, 9, 24, 11, -66, -152, -8, 587, 1082, -438, -5248, -7733, 7942, 47502, 53792, -105313, -430118, -343043, 1249800, 3866557, 1730018, -13996096, -34243896, -1947203, 150962374, 296101865, -121857184, -1582561869, -2468098042, 2529520766
Offset: 0
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a(n) = sum(k=0, n\2, (-1)^k*binomial(n+1, 2*k+1)*binomial(2*k, k)/(k+1));
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my(N=40, x='x+O('x^N)); Vec(2/((1-x)*(1-x+sqrt((1-x)^2+4*x^2))))
A360059
a(n) = Sum_{k=0..n} (-1)^k * binomial(n+3*k+3,n-k) * Catalan(k).
Original entry on oeis.org
1, 3, 4, 3, 5, 12, 6, -13, 29, 95, -130, -304, 895, 1050, -5068, -2181, 27743, -5481, -143532, 117983, 700831, -1074414, -3163138, 7872784, 12585117, -51587107, -38040886, 312988334, 18178883, -1779688404, 1013771196, 9485832411, -11749675733, -46878057651
Offset: 0
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Table[Sum[(-1)^k Binomial[n+3k+3,n-k]CatalanNumber[k],{k,0,n}],{n,0,40}] (* Harvey P. Dale, May 06 2024 *)
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a(n) = sum(k=0, n, (-1)^k*binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));
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my(N=40, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1+4*x/(1-x)^4))))
Showing 1-4 of 4 results.