A365223
G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 + x*A(x)^4).
Original entry on oeis.org
1, 1, 2, 3, -3, -50, -244, -714, -530, 8522, 63548, 259473, 535647, -1321437, -19094684, -103022071, -322370363, -142186810, 5537336460, 41081448638, 170484444654, 332739198585, -1241023311708, -15677607031084, -83737193010368, -255608722098225, -12706843586158
Offset: 0
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a(n) = sum(k=0, n, (-1)^(n-k)*binomial(4*n-k+1, k)*binomial(n-1, n-k)/(4*n-k+1));
A370720
G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/4) / (1+x))^4.
Original entry on oeis.org
1, 4, 14, 56, 241, 1088, 5082, 24352, 119036, 591224, 2975150, 15136036, 77721311, 402276364, 2096572304, 10993229392, 57951531087, 306954017592, 1632807888084, 8719002979360, 46720890435026, 251149205370864, 1353974197346154, 7318852828505148
Offset: 0
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a(n) = 4*sum(k=0, n, (-1)^(n-k)*binomial(n-1, n-k)*binomial(3*k+4, k)/(3*k+4));
A371428
Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 - x^2) ).
Original entry on oeis.org
1, 3, 11, 46, 209, 1003, 5002, 25665, 134605, 718371, 3888633, 21298962, 117823660, 657344600, 3694378463, 20896495211, 118865999117, 679545095167, 3902327585407, 22499738052954, 130200110475407, 755927955655813, 4402088019958400, 25706104810367515
Offset: 0
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Table[1/(n+1) Sum[(-1)^k Binomial[n+1,k]Binomial[3n-3k+3,n-2k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Sep 25 2024 *)
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3-x^2))/x)
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a(n) = sum(k=0, n\2, (-1)^k*binomial(n+1, k)*binomial(3*n-3*k+3, n-2*k))/(n+1);
A371429
Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 - x^4) ).
Original entry on oeis.org
1, 3, 12, 55, 272, 1413, 7599, 41933, 236053, 1350093, 7822620, 45817390, 270815730, 1613300978, 9676131942, 58380176644, 354081959367, 2157570900137, 13201923181308, 81084900544971, 499711105642851, 3089163236655363, 19150916212748940, 119031956868317285
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3-x^4))/x)
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a(n) = sum(k=0, n\4, (-1)^k*binomial(n+1, k)*binomial(3*n-3*k+3, n-4*k))/(n+1);
A377705
G.f. A(x) satisfies A(x) = 1 + x/A(x)^2 * (1 - A(x) + A(x)^3).
Original entry on oeis.org
1, 1, 0, 2, -3, 12, -35, 121, -413, 1464, -5265, 19249, -71236, 266443, -1005511, 3824055, -14641264, 56389272, -218315173, 849170605, -3316817080, 13004273475, -51160638706, 201901154910, -799059730844, 3170706566751, -12611882813645, 50277271079611
Offset: 0
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a(n) = if(n==0, 1, sum(k=0, n, (-1)^k*binomial(n, k)*binomial(n-3*k, n-k-1))/n);