cp's OEIS Frontend

A111845 Triangle P, read by rows, that satisfies [P^4](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(4*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(k,k)=1 and P(k+1,1)=P(k+1,0) for k>=0.

%I A111845 #13 Jul 11 2025 15:34:56
%S A111845 1,1,1,4,4,1,40,40,16,1,1040,1040,544,64,1,78240,78240,48960,8320,256,
%T A111845 1,18504256,18504256,13110400,2878720,131584,1024,1,14463224448,
%U A111845 14463224448,11192599808,2982187520,180270080,2099200,4096,1
%N A111845 Triangle P, read by rows, that satisfies [P^4](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(4*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(k,k)=1 and P(k+1,1)=P(k+1,0) for k>=0.
%C A111845 Column 0 and column 1 are equal for n>0.
%F A111845 Let q=4; 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 = Sum_{n>=1} -(-1)^n/n!*Product_{j=0..n-1} L(q^j*x); L(x) equals the g.f. of column 0 of the matrix log of P (A111849).
%e A111845 Let q=4; the g.f. of column k of matrix power P^m is:
%e A111845 1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +
%e A111845 (m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +
%e A111845 (m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...
%e A111845 where L(x) satisfies:
%e A111845 x = L(x) - L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! +- ...
%e A111845 and L(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4 + ... (A111849).
%e A111845 Thus the g.f. of column 0 of matrix power P^m is:
%e A111845 1 + m*L(x) + m^2/2!*L(x)*L(4*x) + m^3/3!*L(x)*L(4*x)*L(4^2*x) +
%e A111845 m^4/4!*L(x)*L(4*x)*L(4^2*x)*L(4^3*x) + ...
%e A111845 Triangle P begins:
%e A111845 1;
%e A111845 1,1;
%e A111845 4,4,1;
%e A111845 40,40,16,1;
%e A111845 1040,1040,544,64,1;
%e A111845 78240,78240,48960,8320,256,1;
%e A111845 18504256,18504256,13110400,2878720,131584,1024,1; ...
%e A111845 where P^4 shifts columns left and up one place:
%e A111845 1;
%e A111845 4,1;
%e A111845 40,16,1;
%e A111845 1040,544,64,1;
%e A111845 78240,48960,8320,256,1; ...
%o A111845 (PARI) {P(n,k,q=4) = my(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,B[i,j]=1,if(j==1,B[i,j]=(A^q)[i-1,1], B[i,j]=(A^q)[i-1,j-1]));));A=B);return(A[n+1,k+1]))}
%o A111845 \\ Print Triangular Matrix (from _Paul D. Hanna_, Jul 11 2025):
%o A111845 for(n=0,10, for(k=0,n, print1(P(n,k),", ")); print(""))
%Y A111845 Cf. A111846 (column 0), A111847 (row sums), A111848 (matrix log), A111840 (q=3), A078536 (variant).
%K A111845 nonn,tabl
%O A111845 0,4
%A A111845 _Paul D. Hanna_, Aug 23 2005