cp's OEIS Frontend

A118022 Triangle T, read by rows, such that T^2 = SHIFT-UP(T); i.e., the matrix square of T shifts each column of T up 1 row, dropping the main diagonal consisting of the powers of 2: [T^2](n,k) = T(n+1,k) with T(n,n) = 2^n for n>=k>=0.

%I A118022 #4 Mar 30 2012 18:36:56
%S A118022 1,1,2,3,4,4,19,24,16,8,243,304,192,64,16,6227,7776,4864,1536,256,32,
%T A118022 319251,398528,248832,77824,12288,1024,64,32737427,40864128,25505792,
%U A118022 7962624,1245184,98304,4096,128,6714170259,8380781312,5230608384
%N A118022 Triangle T, read by rows, such that T^2 = SHIFT-UP(T); i.e., the matrix square of T shifts each column of T up 1 row, dropping the main diagonal consisting of the powers of 2: [T^2](n,k) = T(n+1,k) with T(n,n) = 2^n for n>=k>=0.
%C A118022 Column 0 is A118023, where T(n,k) = A118023(n-k)*2^(k*(n-k+1)).
%F A118022 G.f. for column k: 2^k = Sum{n>=0} T(n+k,k)*x^n*prod_{j=0..n} (1-2^(j+k)*x). T(n,k) = T(n-k,0)*2^(k*(n-k+1)) = A118023(n-k)*2^(k*(n-k+1)).
%e A118022 Triangle T begins:
%e A118022 1;
%e A118022 1,2;
%e A118022 3,4,4;
%e A118022 19,24,16,8;
%e A118022 243,304,192,64,16;
%e A118022 6227,7776,4864,1536,256,32;
%e A118022 319251,398528,248832,77824,12288,1024,64;
%e A118022 32737427,40864128,25505792,7962624,1245184,98304,4096,128; ...
%e A118022 Matrix square, T^2, equals SHIFT_UP(T):
%e A118022 1;
%e A118022 3,4;
%e A118022 19,24,16;
%e A118022 243,304,192,64;
%e A118022 6227,7776,4864,1536,256;
%e A118022 319251,398528,248832,77824,12288,1024; ...
%e A118022 G.f. for column 0: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-4x) + ...
%e A118022 + T(n,0)*x^n*(1-x)(1-2x)(1-4x)*..*(1-2^n*x) + ...
%e A118022 G.f. for column 1: 2 = 2(1-2x) + 4*x*(1-2x)(1-4x) + 24*x^2*(1-2x)(1-4x)(1-8x) + ...
%e A118022 + T(n+1,1)*x^n*(1-2x)(1-4x)(1-8x)*..*(1-2^(n+1)*x) + ...
%e A118022 G.f. for column 2: 4 = 4(1-4x) + 16*x*(1-4x)(1-8x) + 192*x^2*(1-4x)(1-8x)(1-16x) + ...
%e A118022 + T(n+2,2)*x^n*(1-4x)(1-8x)(1-16x)*..*(1-2^(n+2)*x) + ...
%o A118022 (PARI) {T(n, k)=local(A=matrix(1, 1), B); A[1, 1]=1; for(m=2, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, i]=2^(i-1), if(j==1, B[i, j]=(A^2)[i-1, 1], B[i, j]=(A^2)[i-1, j])); )); A=B); return(A[n+1, k+1])}
%Y A118022 Cf. A118023 (column 0); A117401 (related triangle); A118024 (variant).
%K A118022 nonn,tabl
%O A118022 0,3
%A A118022 _Paul D. Hanna_, Apr 10 2006