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.
%I A244652 #8 Jul 04 2014 11:22:32
%S A244652 1,19,262,3322,41455,520165,6602716,84860884,1103478733,14500102087,
%T A244652 192309166018,2571407785918,34631087423419,469382779109305,
%U A244652 6398055968407480,87653105740545976,1206315271455768505,16669999282643795899,231219555870655381438,3217973871571202211778
%N A244652 G.f. A(x) satisfies: Sum_{k=0..n} [x^k] A(x)^n = binomial(6*n,3*n).
%C A244652 Compare to Sum_{k=0..n} [x^k] 1/(1-x)^(5*n) = binomial(6*n,n).
%C A244652 Compare to Sum_{k=0..n} [x^k] 1/((1-x)*(1-2*x)^2)^n = binomial(4*n,2*n).
%H A244652 Paul D. Hanna, <a href="/A244652/b244652.txt">Table of n, a(n) for n = 0..100</a>
%F A244652 Recurrence: n*(n+2)*(n^2 + 5*n - 2)*a(n) = (18*n^4 + 103*n^3 + 39*n^2 + 56*n + 12)*a(n-1) - (49*n^4 + 264*n^3 - 169*n^2 - 372*n - 180)*a(n-2) + 8*(6*n^4 + 31*n^3 - 51*n^2 - 112*n - 48)*a(n-3) - 16*(n-3)*(n+1)*(n^2 + 7*n + 4)*a(n-4). - _Vaclav Kotesovec_, Jul 04 2014
%F A244652 a(n) ~ 64*sqrt(6*(26*sqrt(3)-45)) * (8+4*sqrt(3))^n / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 04 2014
%e A244652 G.f.: A(x) = 1 + 19*x + 262*x^2 + 3322*x^3 + 41455*x^4 + 520165*x^5 +...
%e A244652 ILLUSTRATION OF INITIAL TERMS.
%e A244652 If we form an array of coefficients of x^k in A(x)^n, n>=0, like so:
%e A244652 A^0: [1], 0, 0, 0, 0, 0, 0, ...;
%e A244652 A^1: [1, 19], 262, 3322, 41455, 520165, 6602716, ...;
%e A244652 A^2: [1, 38, 885], 16600, 277790, 4356348, 65729806, ...;
%e A244652 A^3: [1, 57, 1869, 46693], 992751, 19018983, 339483259, ...;
%e A244652 A^4: [1, 76, 3214, 100460, 2600405], 59206736, 1229790360, ...;
%e A244652 A^5: [1, 95, 4920, 184760, 5645140, 149282604],3549124200, ...;
%e A244652 A^6: [1, 114, 6987, 306452, 10801665, 325750014, 8738270067],...; ...
%e A244652 then we can illustrate how the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals C(6*n,3*n):
%e A244652 C( 0, 0) = 1 = 1;
%e A244652 C( 6, 3) = 1 + 19 = 20;
%e A244652 C(12, 6) = 1 + 38 + 885 = 924;
%e A244652 C(18, 9) = 1 + 57 + 1869 + 46693 = 48620;
%e A244652 C(24,18) = 1 + 76 + 3214 + 100460 + 2600405 = 2704156;
%e A244652 C(30,21) = 1 + 95 + 4920 + 184760 + 5645140 + 149282604 = 155117520;
%e A244652 C(36,24) = 1 + 114 + 6987 + 306452 + 10801665 + 325750014 + 8738270067 = 9075135300; ...
%o A244652 (PARI) /* By Definition (slow): */
%o A244652 {a(n)=if(n==0, 1, ( binomial(6*n,3*n) - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j/1!)^n + x*O(x^k), k)))/n)}
%o A244652 for(n=0, 20, print1(a(n), ", "))
%o A244652 (PARI) /* Faster, using series reversion: */
%o A244652 {a(n)=local(B=sum(k=0, n+1, binomial(6*k,3*k)*x^k)+x^3*O(x^n), G=1+x*O(x^n));
%o A244652 for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); polcoeff(x/serreverse(x*G), n)}
%o A244652 for(n=0, 30, print1(a(n), ", "))
%Y A244652 Cf. A232606, A232683, A232687, A244650, A244651, A244653, A244654.
%K A244652 nonn
%O A244652 0,2
%A A244652 _Paul D. Hanna_, Jul 03 2014