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%I #8 Jun 14 2017 00:28:00
%S 1,1,1,2,2,1,5,6,4,1,16,24,20,8,1,69,136,136,72,16,1,430,1162,1360,
%T 880,272,32,1,4137,15702,21204,16032,6240,1056,64,1,64436,346768,
%U 537748,461992,214336,46784,4160,128,1,1676353,12836904,22891448,21944520
%N Triangle T, read by rows, where column n of T = column 0 of T^(2^n) for n>0, such that column 0 (A129092) equals the row sums of the prior row, starting with T(0,0)=1.
%C T(n,0) = A129092(n) = A030067(2^n - 1) for n>0 where A030067 is the Semi-Fibonacci numbers.
%F Row k = row 0 of matrix power A129104^k, where A129104 equals triangle A129100 with an additional leftmost column of all 1's.
%e Column 0 of row n equals A129092(n) = A030067(2^n-1) for n>=1,
%e where A030067 is the semi-Fibonacci numbers:
%e [(1), 1, (2), 1, 3, 2, (5), 1, 6, 3, 9, 2, 11, 5, (16), 1, ...],
%e which obey the recurrence:
%e A030067(n) = A030067(n/2) when n is even; and
%e A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 5, 6, 4, 1;
%e 16, 24, 20, 8, 1;
%e 69, 136, 136, 72, 16, 1;
%e 430, 1162, 1360, 880, 272, 32, 1;
%e 4137, 15702, 21204, 16032, 6240, 1056, 64, 1;
%e 64436, 346768, 537748, 461992, 214336, 46784, 4160, 128, 1;
%e 1676353, 12836904, 22891448, 21944520, 11720016, 3107456, 361856, 16512, 256, 1; ...
%e where columns shift left under matrix square, A129100^2, which starts:
%e 1;
%e 2, 1;
%e 6, 4, 1;
%e 24, 20, 8, 1;
%e 136, 136, 72, 16, 1;
%e 1162, 1360, 880, 272, 32, 1; ...
%e Inserting a left column of all 1's, yields matrix A129104:
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 5, 6, 4, 1;
%e 1, 16, 24, 20, 8, 1;
%e 1, 69, 136, 136, 72, 16, 1; ...
%e where row 0 of matrix power A129104^k forms row k of A129100,
%e as illustrated below.
%e For row 2: A129104^2 begins:
%e 2, 2, 1;
%e 3, 4, 3, 1;
%e 6, 12, 12, 6, 1;
%e 17, 54, 65, 42, 12, 1;
%e 70, 362, 512, 400, 156, 24, 1;
%e 431, 3708, 6223, 5656, 2744, 600, 48, 1; ...
%e and row 0 of A129104^2 equals row 2 of A129100: [2, 2, 1].
%e For row 3: A129104^3 begins:
%e 5, 6, 4, 1;
%e 11, 18, 16, 7, 1;
%e 37, 88, 96, 56, 14, 1;
%e 191, 672, 860, 609, 210, 28, 1;
%e 1525, 8038, 11956, 9856, 4256, 812, 56, 1; ...
%e and row 0 of A129104^3 equals row 3 of A129100: [5, 6, 4, 1].
%e For row 4: A129104^4 begins:
%e 16, 24, 20, 8, 1;
%e 53, 112, 116, 64, 15, 1;
%e 292, 890, 1088, 736, 240, 30, 1;
%e 2571, 11350, 16056, 12664, 5185, 930, 60, 1; ...
%e and row 0 of A129104^4 equals row 4 of A129100: [16, 24, 20, 8, 1].
%o (PARI) T(n,k)=local(A=Mat(1),B);for(m=1,n+1,B=matrix(m,m);for(r=1,m,for(c=1,r, if(r==c || r==1 || r==2,B[r,c]=1,if(c==1,B[r,1]=sum(i=1,r-1,A[r-1,i]), B[r,c]=(A^(2^(c-1)))[r-c+1,1])); )); A=B); return(A[n+1, k+1])
%Y Cf. A030067 (Semi-Fibonacci); A129092 (row sums=column 0), A129101 (column 1), A129102 (column 2), A129103 (column 3); variant: A129104.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Mar 29 2007
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