%I #7 Jun 13 2017 22:36:47

%S 1,1,1,1,7,1,1,154,49,1,1,16275,8281,343,1,1,9106461,6558209,410914,

%T 2401,1,1,28543862991,27307109501,2298650515,20170801,16807,1,1,

%U 521136519414483,636922972420469,67522139062441,790856748801,988621354

%N Triangle P, read by rows, that satisfies [P^7](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(7*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) = partitions of (7^n - 7^(n-k)) into powers of 7 <= 7^(n-k).

%F Let q=7; 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 (A111834).

%e Let q=7; 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 - 5/2!*x^2 + 83/3!*x^3 + 16110/4!*x^4 +... (A111834).

%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(7*x) + m^3/3!*L(x)*L(7*x)*L(7^2*x) +

%e m^4/4!*L(x)*L(7*x)*L(7^2*x)*L(7^3*x) + ...

%e Triangle P begins:

%e 1;

%e 1,1;

%e 1,7,1;

%e 1,154,49,1;

%e 1,16275,8281,343,1;

%e 1,9106461,6558209,410914,2401,1;

%e 1,28543862991,27307109501,2298650515,20170801,16807,1; ...

%e where P^7 shifts columns left and up one place:

%e 1;

%e 7,1;

%e 154,49,1;

%e 16275,8281,343,1; ...

%o (PARI) P(n,k,q=7)=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. A111831 (column 1), A111832 (row sums), A111833 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111825 (q=6), A111835 (q=8).

%K frac,nonn,tabl

%O 0,5

%A _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005