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.

1, 1, 2, 3, 4, 4, 19, 24, 16, 8, 243, 304, 192, 64, 16, 6227, 7776, 4864, 1536, 256, 32, 319251, 398528, 248832, 77824, 12288, 1024, 64, 32737427, 40864128, 25505792, 7962624, 1245184, 98304, 4096, 128, 6714170259, 8380781312, 5230608384

Offset: 0

Author

Paul D. Hanna, Apr 10 2006

Keywords

• nonn
• tabl

Comments

Column 0 is A118023, where T(n,k) = A118023(n-k)*2^(k*(n-k+1)).

Examples

			Triangle T begins:
1;
1,2;
3,4,4;
19,24,16,8;
243,304,192,64,16;
6227,7776,4864,1536,256,32;
319251,398528,248832,77824,12288,1024,64;
32737427,40864128,25505792,7962624,1245184,98304,4096,128; ...
Matrix square, T^2, equals SHIFT_UP(T):
1;
3,4;
19,24,16;
243,304,192,64;
6227,7776,4864,1536,256;
319251,398528,248832,77824,12288,1024; ...
G.f. for column 0: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-4x) + ...
+ T(n,0)*x^n*(1-x)(1-2x)(1-4x)*..*(1-2^n*x) + ...
G.f. for column 1: 2 = 2(1-2x) + 4*x*(1-2x)(1-4x) + 24*x^2*(1-2x)(1-4x)(1-8x) + ...
+ T(n+1,1)*x^n*(1-2x)(1-4x)(1-8x)*..*(1-2^(n+1)*x) + ...
G.f. for column 2: 4 = 4(1-4x) + 16*x*(1-4x)(1-8x) + 192*x^2*(1-4x)(1-8x)(1-16x) + ...
+ T(n+2,2)*x^n*(1-4x)(1-8x)(1-16x)*..*(1-2^(n+2)*x) + ...
		

Crossrefs

Cf. A118023 (column 0); A117401 (related triangle); A118024 (variant).

Programs

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])}

Formula

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)).