A369504
Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^3)^2 ).
Original entry on oeis.org
1, 4, 22, 142, 1005, 7546, 59033, 475962, 3927204, 33001024, 281449964, 2429922400, 21196031340, 186521336460, 1653830553417, 14761130834428, 132516050272100, 1195778542160992, 10839917478886459, 98671228898404032, 901509955793840923
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x^3)^2)/x)
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a(n) = sum(k=0, n\3, binomial(2*n+2, k)*binomial(4*n-2*k+4, n-3*k))/(n+1);
A370285
Coefficient of x^n in the expansion of ( (1+x)^2 + x^3 )^n.
Original entry on oeis.org
1, 2, 6, 23, 94, 392, 1659, 7107, 30734, 133880, 586576, 2582142, 11411371, 50597900, 224986467, 1002867878, 4479814606, 20049099908, 89878609344, 403521966942, 1814102538624, 8165526187128, 36794746597494, 165968135843522, 749314496125451, 3385881647958442
Offset: 0
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a := n -> binomial(2*n, n) * hypergeom([(1-n)/3, (2-n)/3, -n/3], [1/2-n, n+1], 27/4):
seq(simplify(a(n)), n = 0..25); # Peter Luschny, Jan 04 2025
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a(n) = sum(k=0, n\3, binomial(n, k)*binomial(2*n-2*k, n-3*k));
A371426
Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 - x^3) ).
Original entry on oeis.org
1, 2, 5, 13, 34, 87, 212, 471, 858, 740, -3674, -29291, -141951, -576379, -2111677, -7161898, -22646026, -66408560, -176815194, -403468266, -641064024, 337909918, 9269952852, 55908644837, 256989808831, 1033152002312, 3792152422259, 12903091079930, 40749582818221
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2-x^3))/x)
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a(n) = sum(k=0, n\3, (-1)^k*binomial(n+1, k)*binomial(2*n-2*k+2, n-3*k))/(n+1);
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