A366366
G.f. satisfies A(x) = (1 + x/A(x)^4)/(1 - x).
Original entry on oeis.org
1, 2, -6, 58, -574, 6402, -75878, 939290, -12000318, 157050178, -2094657926, 28368411194, -389079656446, 5393118559938, -75431624084838, 1063251390845338, -15088643098754942, 215396586102923138, -3091050571516120582, 44566089825496186170
Offset: 0
-
a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k-1, k)*binomial(4*k-1, n-k)/(5*k-1));
A378237
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(n,r) * binomial(n+3*r+k,n)/(n+3*r+k) for k > 0.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 1, 4, 10, 0, 1, 6, 24, 74, 0, 1, 8, 42, 188, 642, 0, 1, 10, 64, 350, 1680, 6082, 0, 1, 12, 90, 568, 3234, 16212, 60970, 0, 1, 14, 120, 850, 5440, 31878, 164584, 635818, 0, 1, 16, 154, 1204, 8450, 54888, 328426, 1732172, 6826690, 0, 1, 18, 192, 1638, 12432, 87402, 574848, 3494142, 18728352, 74958914, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, 12, ...
0, 10, 24, 42, 64, 90, 120, ...
0, 74, 188, 350, 568, 850, 1204, ...
0, 642, 1680, 3234, 5440, 8450, 12432, ...
0, 6082, 16212, 31878, 54888, 87402, 131964, ...
0, 60970, 164584, 328426, 574848, 931770, 1433544, ...
-
T(n, k, t=1, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))
A348957
G.f. A(x) satisfies A(x) = (1 + x * A(-x)) / (1 - x * A(x)).
Original entry on oeis.org
1, 2, 2, 10, 18, 98, 210, 1194, 2786, 16258, 39906, 236938, 601458, 3615330, 9399858, 57024426, 150947010, 922283522, 2475603138, 15212318730, 41290579410, 254909413218, 698230131858, 4327273358250, 11943274468770, 74260741616514, 206279837823650, 1286199407132554
Offset: 0
-
nmax = 27; A[] = 0; Do[A[x] = (1 + x A[-x])/(1 - x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -(-1)^n a[n - 1] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 27}]
CoefficientList[y/.AsymptoticSolve[y-y^2+x(1+y^3)==0,y->1,{x,0,27}][[1]],x] (* Alexander Burstein, Nov 26 2021 *)
A364195
Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^5 * (1 + A(x)^2).
Original entry on oeis.org
1, 2, 24, 412, 8280, 181904, 4232048, 102479184, 2555884896, 65207430848, 1693785940992, 44643489969792, 1190986788639232, 32097745138518528, 872595854798515456, 23900545715576753408, 658934625866433496576, 18271554709525993556992, 509241947434834351042560
Offset: 0
-
a(n) = sum(k=0, n, binomial(n, k)*binomial(5*n+2*k+1, n)/(5*n+2*k+1));
A376160
G.f. satisfies A(x) = 1 / ((1-x)^3 - x*A(x)^3).
Original entry on oeis.org
1, 4, 25, 260, 3205, 42966, 609567, 8999164, 136811781, 2127343669, 33675622992, 540878965522, 8792433396559, 144383416380703, 2391557494237062, 39910530610590312, 670383542665237001, 11325278943044058378, 192301381444863249559, 3280101940070399446926
Offset: 0
-
a(n) = sum(k=0, n, binomial(n+11*k+2, n-k)*binomial(4*k, k)/(3*k+1));
A379194
G.f. A(x) satisfies A(x) = (1 + x*A(x))^2/(1 - x*A(x)^3).
Original entry on oeis.org
1, 3, 19, 174, 1883, 22323, 280409, 3666736, 49386326, 680431419, 9544684113, 135852904486, 1957119390279, 28482417043498, 418119577938769, 6184065626127498, 92062362629472668, 1378427894172778961, 20744229318047760620, 313606289763390553200, 4760422971894347226659
Offset: 0
-
a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(2*n+4*k+2, n-k)/(n+3*k+1));