A346735
G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x)^3.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 3, 6, 10, 15, 21, 34, 63, 120, 220, 381, 642, 1102, 1968, 3615, 6658, 12090, 21675, 38820, 70200, 128466, 236583, 435453, 798798, 1462933, 2684352, 4945740, 9145839, 16942356, 31388571, 58140726, 107753364, 199993359, 371852269, 692375844, 1290252474
Offset: 0
-
nmax = 40; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 5, 1, Sum[Sum[a[i] a[j] a[n - i - j - 5], {j, 0, n - i - 5}], {i, 0, n - 5}]]; Table[a[n], {n, 0, 40}]
A366590
G.f. A(x) satisfies A(x) = 1 + x^2*(1+x)^2*A(x)^3.
Original entry on oeis.org
1, 0, 1, 2, 4, 12, 30, 84, 238, 680, 1993, 5882, 17575, 52976, 160870, 491924, 1512940, 4677672, 14529744, 45320640, 141897039, 445792908, 1404899598, 4440113940, 14069493813, 44689897200, 142268117566, 453839997836, 1450547245960, 4644492976232, 14896047099592
Offset: 0
-
a(n) = sum(k=0, n\2, binomial(2*k, n-2*k)*binomial(3*k, k)/(2*k+1));
A378321
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3*n-3*r+k,r) * binomial(r,n-r)/(3*n-3*r+k) for k > 0.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 8, 6, 0, 1, 5, 10, 16, 19, 16, 0, 1, 6, 15, 28, 42, 50, 42, 0, 1, 7, 21, 45, 79, 114, 137, 114, 0, 1, 8, 28, 68, 135, 224, 322, 380, 322, 0, 1, 9, 36, 98, 216, 401, 652, 918, 1088, 918, 0, 1, 10, 45, 136, 329, 672, 1205, 1912, 2673, 3152, 2673, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 3, 6, 10, 15, 21, ...
0, 3, 8, 16, 28, 45, 68, ...
0, 6, 19, 42, 79, 135, 216, ...
0, 16, 50, 114, 224, 401, 672, ...
0, 42, 137, 322, 652, 1205, 2088, ...
-
T(n, k, t=0, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))
A367260
G.f. satisfies A(x) = 1 + x*A(x)^3 * (1 + x*A(x))^3.
Original entry on oeis.org
1, 1, 6, 36, 251, 1881, 14817, 120950, 1014042, 8680377, 75552553, 666614637, 5948817600, 53599239101, 486926148000, 4455202562652, 41018936164660, 379747493741643, 3532914858433284, 33012260400580342, 309692626084981245, 2915659701275923491
Offset: 0
-
a(n, s=3, t=3, u=1) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));
A378411
G.f. A(x) satisfies A(x) = ( 1 + x * (1 + x*A(x)^(3/2)) )^2.
Original entry on oeis.org
1, 2, 3, 8, 19, 50, 137, 380, 1088, 3152, 9270, 27576, 82794, 250700, 764454, 2345688, 7237318, 22438988, 69876356, 218456216, 685400835, 2157396738, 6810801959, 21559694364, 68417766207, 217617573110, 693655532081, 2215401956720, 7088605614314, 22720370822508
Offset: 0
-
a(n, r=2, s=1, t=0, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));