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Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^2)^2 ).
+10
3
1, 4, 24, 168, 1284, 10384, 87360, 756704, 6703168, 60444928, 552990592, 5120101760, 47887472000, 451759449600, 4293634467840, 41073654689280, 395170166443008, 3821262491103232, 37118973530660864, 362035991963869184, 3544080121528001536
FORMULA
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+2,k) * binomial(4*n-2*k+4,n-2*k).
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x^2)^2)/x)
(PARI) a(n) = sum(k=0, n\2, binomial(2*n+2, k)*binomial(4*n-2*k+4, n-2*k))/(n+1);
Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^3)^2 ).
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3
1, 4, 22, 142, 1005, 7546, 59033, 475962, 3927204, 33001024, 281449964, 2429922400, 21196031340, 186521336460, 1653830553417, 14761130834428, 132516050272100, 1195778542160992, 10839917478886459, 98671228898404032, 901509955793840923
FORMULA
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+2,k) * binomial(4*n-2*k+4,n-3*k).
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x^3)^2)/x)
(PARI) a(n) = sum(k=0, n\3, binomial(2*n+2, k)*binomial(4*n-2*k+4, n-3*k))/(n+1);
Expansion of (1/x) * Series_Reversion( x / ((1+x)^3+x)^2 ).
+10
1
1, 8, 86, 1066, 14361, 204314, 3020745, 45955442, 714723588, 11312450432, 181625888244, 2950848879096, 48423670556100, 801454908292020, 13363137183238881, 224253208102065664, 3784736105491395780, 64197997357038408976, 1093863031541592651003
FORMULA
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(2*n+2,k) * binomial(6*n-3*k+6,n-k).
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3+x)^2)/x)
(PARI) a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(6*n-3*k+6, n-k))/(n+1);
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