A378958
G.f. A(x) satisfies A(x) = 1 + x/A(x) * (1 - A(x) + A(x)^4).
Original entry on oeis.org
1, 1, 2, 8, 32, 145, 681, 3337, 16773, 86181, 450268, 2385544, 12784861, 69189509, 377576512, 2075423744, 11480230037, 63857579629, 356962271136, 2004255583560, 11298268724556, 63919517790933, 362806671879955, 2065443363987045, 11790688867079872, 67477283970889867
Offset: 0
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a(n) = if(n==0, 1, sum(k=0, n, (-1)^k*binomial(n, k)*binomial(3*n-4*k, n-k-1))/n);
A383119
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(4*k,k).
Original entry on oeis.org
1, 3, 21, 147, 1093, 8343, 64869, 510891, 4062277, 32539647, 262181601, 2122581123, 17252278789, 140695104943, 1150670390541, 9433965332127, 77512716483461, 638080242074447, 5261486780929209, 43450477494413751, 359308411992366513, 2974886601163646379, 24657831769475675253
Offset: 0
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Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[4 k, k], {k, 0, n}], {n, 0, 22}]
Table[(-1)^n HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {1/3, 2/3, 1}, 256/27], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(1/x) Sum[Binomial[4 k, k] (x/(1 + x))^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
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a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(4*k, k)); \\ Seiichi Manyama, Apr 17 2025
A371430
Expansion of (1/x) * Series_Reversion( x / ((1+x)^4 - x^2) ).
Original entry on oeis.org
1, 4, 21, 128, 851, 5984, 43759, 329396, 2535406, 19863592, 157874971, 1269833668, 10316765299, 84540929568, 697928139977, 5799156785376, 48461097907978, 407020852551016, 3434002483872566, 29090171931564848, 247333930963224287, 2109921586071433064
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^4-x^2))/x)
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a(n) = sum(k=0, n\2, (-1)^k*binomial(n+1, k)*binomial(4*n-4*k+4, n-2*k))/(n+1);
A371545
G.f. A(x) satisfies A(x) = (1 + x*A(x)^2 / (1+x))^2.
Original entry on oeis.org
1, 2, 7, 36, 209, 1312, 8663, 59298, 416961, 2993790, 21857208, 161768154, 1210949944, 9152366596, 69745701746, 535304423948, 4134240993874, 32105769989714, 250551371644825, 1963871015580984, 15454014753626079, 122044659649929546, 966951068210379441
Offset: 0
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a(n) = 2*sum(k=0, n, (-1)^(n-k)*binomial(n-1, n-k)*binomial(4*k+2, k)/(4*k+2));
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