cp's OEIS Frontend

A111820 Triangle P, read by rows, that satisfies [P^5](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(5*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.

%I A111820 #7 Jun 13 2017 22:34:21
%S A111820 1,1,1,1,5,1,1,55,25,1,1,2055,1525,125,1,1,291430,311525,38875,625,1,
%T A111820 1,165397680,239305275,40338875,975625,3125,1,1,390075741430,
%U A111820 735920617775,157056792000,5077475625,24409375,15625,1
%N A111820 Triangle P, read by rows, that satisfies [P^5](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(5*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 A111820 Also P(n,k) = the partitions of (5^n - 5^(n-k)) into powers of 5 <= 5^(n-k).
%F A111820 Let q=5; 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 (A111824).
%e A111820 Let q=5; the g.f. of column k of matrix power P^m is:
%e A111820 1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +
%e A111820 (m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +
%e A111820 (m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...
%e A111820 where L(x) satisfies:
%e A111820 x/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...
%e A111820 and L(x) = x - 3/2!*x^2 + 16/3!*x^3 + 2814/4!*x^4 +... (A111824).
%e A111820 Thus the g.f. of column 0 of matrix power P^m is:
%e A111820 1 + m*L(x) + m^2/2!*L(x)*L(5*x) + m^3/3!*L(x)*L(5*x)*L(5^2*x) +
%e A111820 m^4/4!*L(x)*L(5*x)*L(5^2*x)*L(5^3*x) + ...
%e A111820 Triangle P begins:
%e A111820 1;
%e A111820 1,1;
%e A111820 1,5,1;
%e A111820 1,55,25,1;
%e A111820 1,2055,1525,125,1;
%e A111820 1,291430,311525,38875,625,1;
%e A111820 1,165397680,239305275,40338875,975625,3125,1; ...
%e A111820 where P^5 shifts columns left and up one place:
%e A111820 1;
%e A111820 5,1;
%e A111820 55,25,1;
%e A111820 2055,1525,125,1;
%e A111820 291430,311525,38875,625,1; ...
%o A111820 (PARI) P(n,k,q=5)=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 A111820 Cf. A111821 (column 1), A111822 (row sums), A111823 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111825 (q=6), A111830 (q=7), A111835 (q=8).
%K A111820 nonn,tabl
%O A111820 0,5
%A A111820 _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005