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A209424 Triangle defined by g.f.: A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^n * y^k ), as read by rows.

%I A209424 #10 Jan 09 2018 18:06:22
%S A209424 1,1,1,1,3,1,1,12,12,1,1,76,347,76,1,1,701,20429,20429,701,1,1,8477,
%T A209424 1919660,10707908,1919660,8477,1,1,126126,259227625,9203978774,
%U A209424 9203978774,259227625,126126,1,1,2223278,47484618291,12099129236936,72078431500368
%N A209424 Triangle defined by g.f.: A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^n * y^k ), as read by rows.
%C A209424 Column 1 is A060946.
%C A209424 Column 2 is A209425.
%C A209424 Row sums equal A167007.
%C A209424 Antidiagonal sums equal A166894.
%C A209424 Central terms form A209426.
%H A209424 Paul D. Hanna, <a href="/A209424/b209424.txt">Table of n, a(n) for n = 0..495 for Rows 0..30 of this triangle in flattened form.</a>
%e A209424 This triangle begins:
%e A209424 1;
%e A209424 1, 1;
%e A209424 1, 3, 1;
%e A209424 1, 12, 12, 1;
%e A209424 1, 76, 347, 76, 1;
%e A209424 1, 701, 20429, 20429, 701, 1;
%e A209424 1, 8477, 1919660, 10707908, 1919660, 8477, 1;
%e A209424 1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1;
%e A209424 1, 2223278, 47484618291, 12099129236936, 72078431500368, 12099129236936, 47484618291, 2223278, 1; ...
%e A209424 G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+12*y+12*y^2+y^3)*x^3 + (1+76*y+20429*y^2+76*y^3+y^4)*x^4 +...
%e A209424 The logarithm of the g.f. equals the series:
%e A209424 log(A(x,y)) = (1 + y)*x
%e A209424 + (1 + 2^2*y + y^2)*x^2/2
%e A209424 + (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3
%e A209424 + (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4
%e A209424 + (1 + 5^5*y + 10^5*y^2 + 10^5*y^3 + 5^5*y^4 + y^5)*x^5/5 +...
%e A209424 in which the coefficients are found in triangle A209427.
%o A209424 (PARI) {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m,k)^m*y^k))+x*O(x^n)),n,x),k,y)}
%o A209424 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A209424 Cf. A060946, A209425, A167007, A166894, A209426, A209427, A209196.
%K A209424 nonn,tabl
%O A209424 0,5
%A A209424 _Paul D. Hanna_, Mar 08 2012