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 A244651 #12 Jul 04 2014 07:19:43
%S A244651 1,14,135,1148,9325,74634,596083,4775288,38447961,311305350,
%T A244651 2534757855,20749571316,170705908421,1410874891522,11710273480395,
%U A244651 97573698950384,815919118022833,6845174820882174,57601263531202871,486057767175907180,4112073577799441181,34871360280503319674
%N A244651 G.f. A(x) satisfies: Sum_{k=0..n} [x^k] A(x)^n = binomial(6*n,2*n).
%C A244651 Compare to Sum_{k=0..n} [x^k] 1/(1-x)^(5*n) = binomial(6*n,n).
%C A244651 Compare to Sum_{k=0..n} [x^k] 1/((1-x)*(1-2*x)^2)^n = binomial(4*n,2*n).
%H A244651 Paul D. Hanna, <a href="/A244651/b244651.txt">Table of n, a(n) for n = 0..100</a>
%F A244651 Recurrence: n*(n+3)*(4*n+1)*a(n) = (44*n^3 + 91*n^2 + 99*n + 46)*a(n-1) - (76*n^3 + 159*n^2 + 53*n - 60)*a(n-2) + 9*(n-1)*(n+2)*(4*n+5)*a(n-3). - _Vaclav Kotesovec_, Jul 04 2014
%F A244651 a(n) ~ 9^(n+4) / (64*sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 04 2014
%e A244651 G.f.: A(x) = 1 + 14*x + 135*x^2 + 1148*x^3 + 9325*x^4 + 74634*x^5 +...
%e A244651 ILLUSTRATION OF INITIAL TERMS.
%e A244651 If we form an array of coefficients of x^k in A(x)^n, n>=0, like so:
%e A244651 A^0: [1], 0, 0, 0, 0, 0, 0, ...;
%e A244651 A^1: [1, 14], 135, 1148, 9325, 74634, 596083, ...;
%e A244651 A^2: [1, 28, 466], 6076, 69019, 720328, 7117572, ...;
%e A244651 A^3: [1, 42, 993, 17528], 258462, 3377556, 40526262, ...;
%e A244651 A^4: [1, 56, 1716, 38248, 695450, 10968552, 155816996, ...;
%e A244651 A^5: [1, 70, 2635, 70980, 1536195, 28435134, 467948465, ...;
%e A244651 A^6: [1, 84, 3750, 118468, 2975325, 63276528, 1185303544],...; ...
%e A244651 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,2*n):
%e A244651 C( 0, 0) = 1 = 1;
%e A244651 C( 6, 2) = 1 + 14 = 15;
%e A244651 C(12, 4) = 1 + 28 + 466 = 495;
%e A244651 C(18, 6) = 1 + 42 + 993 + 17528 = 18564;
%e A244651 C(24, 8) = 1 + 56 + 1716 + 38248 + 695450 = 735471;
%e A244651 C(30,10) = 1 + 70 + 2635 + 70980 + 1536195 + 28435134 = 30045015;
%e A244651 C(36,12) = 1 + 84 + 3750 + 118468 + 2975325 + 63276528 + 1185303544 = 1251677700; ...
%o A244651 (PARI) /* By Definition (slow): */
%o A244651 {a(n)=if(n==0, 1, ( binomial(6*n,2*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 A244651 for(n=0, 20, print1(a(n), ", "))
%o A244651 (PARI) /* Faster, using series reversion: */
%o A244651 {a(n)=local(B=sum(k=0, n+1, binomial(6*k,2*k)*x^k)+x^3*O(x^n), G=1+x*O(x^n));
%o A244651 for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); polcoeff(x/serreverse(x*G), n)}
%o A244651 for(n=0, 30, print1(a(n), ", "))
%Y A244651 Cf. A232606, A232683, A232687, A244650, A244652, A244653, A244654.
%K A244651 nonn
%O A244651 0,2
%A A244651 _Paul D. Hanna_, Jul 03 2014