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%I #7 Jun 13 2017 22:34:14
%S 1,1,1,1,6,1,1,96,36,1,1,6306,3816,216,1,1,1883076,1625436,139536,
%T 1296,1,1,2700393702,3121837776,360839016,5036256,7776,1,1,
%U 19324893252552,28794284803908,4200503990976,78293629296,181382976,46656,1
%N Triangle P, read by rows, that satisfies [P^6](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(6*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(0,k)=1 and P(k,k)=1 for all k>=0.
%C Also P(n,k) = the partitions of (6^n - 6^(n-k)) into powers of 6 <= 6^(n-k).
%F Let q=6; the g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} L(q^j*x) where L(x) satisfies: x/(1-x) = Sum_{n>=1} Product_{j=0..n-1} L(q^j*x)/(j+1) and L(x) equals the g.f. of column 0 of the matrix log of P (A111829).
%e Let q=6; the g.f. of column k of matrix power P^m is:
%e 1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +
%e (m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +
%e (m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...
%e where L(x) satisfies:
%e x/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...
%e and L(x) = x - 4/2!*x^2 + 42/3!*x^3 + 7296/4!*x^4 +... (A111829).
%e Thus the g.f. of column 0 of matrix power P^m is:
%e 1 + m*L(x) + m^2/2!*L(x)*L(6*x) + m^3/3!*L(x)*L(6*x)*L(6^2*x) +
%e m^4/4!*L(x)*L(6*x)*L(6^2*x)*L(6^3*x) + ...
%e Triangle P begins:
%e 1;
%e 1,1;
%e 1,6,1;
%e 1,96,36,1;
%e 1,6306,3816,216,1;
%e 1,1883076,1625436,139536,1296,1;
%e 1,2700393702,3121837776,360839016,5036256,7776,1; ...
%e where P^6 shifts columns left and up one place:
%e 1;
%e 6,1;
%e 96,36,1;
%e 6306,3816,216,1; ...
%o (PARI) P(n,k,q=6)=local(A=Mat(1),B);if(n<k || k<0,0, for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(A[n+1,k+1]))
%Y Cf. A111826 (column 1), A111827 (row sums), A111828 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111830 (q=7), A111835 (q=8).
%K nonn,tabl
%O 0,5
%A _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005
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