This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^3 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
{1},
{1, 1},
{1, 2},
{1, 3},
{1, 4},
{1, 5},
{1, 6},
{1, 7},
{1, 8, 1},
{1, 9, 9},
{1, 10, 45},
{1, 11, 165},
{1, 12, 495},
{1, 13, 1287},
{1, 14, 3003},
{1, 15, 6435},
{1, 16, 12870},
{1, 17, 24310},
{1, 18, 43758},
{1, 19, 75582},
{1, 20, 125970},
...
{1, 27, 2220075, 1}
t[n_, m_] = Binomial[n, m^3];
Table[Table[t[n, m], {m, 0, Floor[n^(1/3)]}], {n, 0, 20}];
Flatten[%]
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 61*x^4 + 306*x^5 + 1623*x^6 +... where the logarithm of the g.f. A = A(x) equals the series: log(A(x)) = (1 + x*A)*x*A + (1 + 2^3*x*A + x^2*A^2)*x^2*A^2/2 + (1 + 3^3*x*A + 3^3*x^2*A^2 + x^3*A^3)*x^3*A^3/3 + (1 + 4^3*x*A + 6^3*x^2*A^2 + 4^3*x^3*A^3 + x^4*A^4)*x^4*A^4/4 + (1 + 5^3*x*A + 10^3*x^2*A^2 + 10^3*x^3*A^3 + 5^3*x^4*A^4 + x^5*A^5)*x^5*A^5/5 + (1 + 6^3*x*A + 15^3*x^2*A^2 + 20^3*x^3*A^3 + 15^3*x^4*A^4 + 6^3*x^5*A^5 + x^6*A^6)*x^6*A^6/6 +... more explicitly, log(A(x)) = x + 5*x^2/2 + 31*x^3/3 + 185*x^4/4 + 1126*x^5/5 + 7043*x^6/6 + 44689*x^7/7 + 286241*x^8/8 +...
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j*A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
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