cp's OEIS Frontend

A218116 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6 * y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.

%I A218116 #5 Oct 21 2012 15:02:17
%S A218116 1,1,1,1,33,1,1,276,276,1,1,1300,12695,1300,1,1,4425,221495,221495,
%T A218116 4425,1,1,12201,2185350,11534720,2185350,12201,1,1,29008,14794261,
%U A218116 285715550,285715550,14794261,29008,1,1,61776,76579851,4276969276,15781532964
%N A218116 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6 * y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.
%C A218116 Compare g.f. to that of the following triangle variants:
%C A218116 * Pascal's: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)*y^k] * x^n/n );
%C A218116 * Narayana: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n );
%C A218116 * A181143: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*y^k] * x^n/n );
%C A218116 * A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*y^k] * x^n/n );
%C A218116 * A218115: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5*y^k] * x^n/n ).
%e A218116 G.f.: A(x,y) = 1 + (1+y)*x + (1+33*y+y^2)*x^2 + (1+276*y+276*y^2+y^3)*x^3 + (1+1300*y+12695*y^2+1300*y^3+y^4)*x^4 +...
%e A218116 The logarithm of the g.f. equals the series:
%e A218116 log(A(x,y)) = (1 + y)*x
%e A218116 + (1 + 2^6*y + y^2)*x^2/2
%e A218116 + (1 + 3^6*y + 3^6*y^2 + y^3)*x^3/3
%e A218116 + (1 + 4^6*y + 6^6*y^2 + 4^6*y^3 + y^4)*x^4/4
%e A218116 + (1 + 5^6*y + 10^6*y^2 + 10^6*y^3 + 5^6*y^4 + y^5)*x^5/5 +...
%e A218116 Triangle begins:
%e A218116 1;
%e A218116 1, 1;
%e A218116 1, 33, 1;
%e A218116 1, 276, 276, 1;
%e A218116 1, 1300, 12695, 1300, 1;
%e A218116 1, 4425, 221495, 221495, 4425, 1;
%e A218116 1, 12201, 2185350, 11534720, 2185350, 12201, 1;
%e A218116 1, 29008, 14794261, 285715550, 285715550, 14794261, 29008, 1;
%e A218116 1, 61776, 76579851, 4276969276, 15781532964, 4276969276, 76579851, 61776, 1;
%e A218116 1, 120825, 324104715, 44480357175, 478591541712, 478591541712, 44480357175, 324104715, 120825, 1; ...
%e A218116 Note that column 1 forms the sum of fifth powers (A000539).
%o A218116 (PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^6*y^j)*x^m/m)+O(x^(n+1))), n, x), k, y)}
%o A218116 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%Y A218116 Cf. A000539 (column 1), A218119 (row sums).
%Y A218116 Cf. variants: A001263 (Narayana), A181143, A181144, A218115.
%K A218116 nonn,tabl
%O A218116 0,5
%A A218116 _Paul D. Hanna_, Oct 21 2012