A364742
G.f. satisfies A(x) = 1 / (1 - x*(1 + x*A(x))^3).
Original entry on oeis.org
1, 1, 4, 13, 50, 201, 841, 3627, 15993, 71803, 327082, 1508002, 7023446, 32995626, 156173668, 744029238, 3565030063, 17169013899, 83061503584, 403483653745, 1967217524551, 9623463731721, 47220968518786, 232354408276613, 1146254897566224, 5668118931395946
Offset: 0
-
a(n) = sum(k=0, n, binomial(n+1, k)*binomial(3*k, n-k))/(n+1);
A371612
G.f. satisfies A(x) = ( 1 + x / (1 - x*A(x)^2) )^2.
Original entry on oeis.org
1, 2, 3, 12, 49, 218, 1037, 5106, 25909, 134410, 709691, 3801498, 20606654, 112828202, 623087675, 3466539248, 19411070496, 109313442562, 618713495451, 3517737628368, 20081523836403, 115058714898196, 661432784830204, 3813891082337178, 22052422636145522
Offset: 0
-
a(n, r=2, s=1, t=0, u=4) = 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));
A364744
G.f. satisfies A(x) = 1 / (1 - x*(1 + x*A(x))^5).
Original entry on oeis.org
1, 1, 6, 26, 131, 706, 3932, 22618, 133099, 797545, 4850296, 29859028, 185712831, 1165227025, 7366475715, 46877977451, 300049605259, 1930395961235, 12476394685445, 80968876247330, 527424073700966, 3447190219684125, 22599794010813360
Offset: 0
-
a(n) = sum(k=0, n, binomial(n+1, k)*binomial(5*k, n-k))/(n+1);
A364762
G.f. satisfies A(x) = 1 / (1 + x*(1 + x*A(x))^4).
Original entry on oeis.org
1, -1, -3, 5, 29, -42, -349, 384, 4705, -3307, -67530, 19392, 1006479, 140594, -15356600, -8897336, 237691865, 246737931, -3708348277, -5655844305, 58027927950, 119178376245, -906834380800, -2396063640645, 14094956420555, 46748815762429, -216921227330074
Offset: 0
-
A364762 := proc(n)
add( (-1)^k*binomial(n+1,k) * binomial(4*k,n-k),k=0..n) ;
%/(n+1) ;
end proc:
seq(A364762(n),n=0..80); # R. J. Mathar, Aug 10 2023
-
nmax = 26; A[_] = 1;
Do[A[x_] = 1/(1 + x*(1 + x*A[x])^4) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 25 2023 *)
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a(n) = sum(k=0, n, (-1)^k*binomial(n+1, k)*binomial(4*k, n-k))/(n+1);
A378731
G.f. A(x) satisfies A(x) = ( 1 + x / (1 - x*A(x)^(4/3)) )^3.
Original entry on oeis.org
1, 3, 6, 22, 93, 417, 1993, 9864, 50217, 261239, 1382448, 7418877, 40278175, 220830513, 1220930337, 6799458685, 38107621704, 214771481163, 1216457185122, 6920603372448, 39529745832681, 226605757331904, 1303291125124071, 7518151040142000, 43488151271999326
Offset: 0
-
a(n, r=3, s=1, t=0, u=4) = 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));
A378732
G.f. A(x) satisfies A(x) = ( 1 + x / (1 - x*A(x)) )^4.
Original entry on oeis.org
1, 4, 10, 36, 155, 704, 3384, 16844, 86097, 449344, 2384170, 12822556, 69743953, 382982940, 2120323014, 11822279232, 66327376437, 374162700460, 2120999728610, 12075668658000, 69021358842795, 395909382981572, 2278286453089574, 13149207655326372, 76096242994616990
Offset: 0
-
a(n, r=4, s=1, t=0, u=4) = 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));
Showing 1-6 of 6 results.