A371518
G.f. A(x) satisfies A(x) = (1 + x*A(x)^2 / (1-x))^2.
Original entry on oeis.org
1, 2, 11, 72, 525, 4104, 33647, 285526, 2486809, 22103726, 199697284, 1828472914, 16929944932, 158246198836, 1491210732346, 14151603542612, 135130396860130, 1297381593071890, 12516650939119421, 121281286192026308, 1179769340479567499
Offset: 0
-
a(n) = 2*sum(k=0, n, binomial(n-1, n-k)*binomial(4*k+1, k)/(3*k+2));
A364792
G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)^2).
Original entry on oeis.org
1, 1, 5, 33, 250, 2054, 17800, 160183, 1482535, 14022415, 134943095, 1317046306, 13005842030, 129708875695, 1304588594925, 13217663310305, 134775670244250, 1382019265706377, 14242560597119165, 147435736533094415, 1532365596794307010
Offset: 0
-
A364792 := proc(n)
if n = 0 then
1;
else
add( binomial(n,k) * binomial(4*n-2*k,n-1-k),k=0..n-1) ;
%/n ;
end if ;
end proc:
seq(A364792(n),n=0..80); # R. J. Mathar, Aug 10 2023
-
a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(4*n-2*k, n-1-k))/n);
A371890
G.f. A(x) satisfies A(x) = 1 - x/A(x)^3 * (1 - A(x) - A(x)^4).
Original entry on oeis.org
1, 1, 2, 1, -4, 0, 37, 16, -313, -214, 3005, 2943, -30391, -39432, 318606, 522863, -3418205, -6889626, 37219105, 90415336, -408758113, -1183054415, 4505089166, 15442590040, -49599878555, -201138280510, 542949788652, 2614332298108, -5877502079248
Offset: 0
-
a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*binomial(n-4*k, n-k-1))/n);
A378325
G.f. A(x) = Sum_{n>=0} a(n)*x^n, where a(n) = Sum_{k=0..n-1} [x^k] A(x)^k for n >= 1 with a(0) = 1.
Original entry on oeis.org
1, 1, 2, 7, 41, 338, 3499, 42969, 606351, 9633640, 169888025, 3290314970, 69409429043, 1584105116525, 38894316619948, 1022411500472240, 28653072049382809, 852911635849385778, 26876978490909421289, 893929164892155754432, 31296785296935394097351, 1150551256823546563078988
Offset: 0
-
{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^k, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ after Paul D. Hanna
A371891
G.f. A(x) satisfies A(x) = 1 - x/A(x)^2 * (1 - A(x) - A(x)^4).
Original entry on oeis.org
1, 1, 3, 8, 21, 61, 203, 724, 2600, 9291, 33525, 123537, 463796, 1759184, 6706976, 25696524, 99069838, 384429159, 1499778661, 5875513183, 23099489574, 91123553946, 360649997698, 1431724692900, 5699142280127, 22741352276386, 90949212893978
Offset: 0
-
a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*binomial(2*n-4*k, n-k-1))/n);
A382916
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^3 / (1-x)^2 ).
Original entry on oeis.org
1, 1, 6, 41, 316, 2636, 23192, 211926, 1992032, 19138016, 187091252, 1855104372, 18612229836, 188601299149, 1927443803738, 19843158497163, 205602235405524, 2142401581747657, 22436439910929038, 236023405797017891, 2492914862240934612, 26426682321857813417
Offset: 0
-
a(n, r=1, s=2, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));
A382917
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^3 / (1-x)^3 ).
Original entry on oeis.org
1, 1, 7, 52, 432, 3878, 36694, 360498, 3642534, 37613947, 395204413, 4211469308, 45409525116, 494500127617, 5430864937915, 60083846523038, 669005596426438, 7491245872785003, 84305386452532885, 953020276395635246, 10816782722212619970, 123218274878407738497
Offset: 0
-
a(n, r=1, s=3, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));
A369213
Expansion of (1/x) * Series_Reversion( x / ((1+x)^4+x^2) ).
Original entry on oeis.org
1, 4, 23, 152, 1091, 8264, 65021, 526236, 4352942, 36637576, 312763225, 2701521420, 23567184019, 207343098824, 1837623853627, 16391011930424, 147029997389386, 1325506554640872, 12003342144724338, 109136630802023808, 995907341988015935
Offset: 0
-
my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^4+x^2))/x)
-
a(n) = sum(k=0, n\2, binomial(n+1, k)*binomial(4*n-4*k+4, n-2*k))/(n+1);