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A132625 Triangle T, read by rows, where row n+1 of T = row n of T^(2^n) with appended '1' for n>=0 with T(0,0)=1.

%I A132625 #7 Aug 09 2015 23:35:46
%S A132625 1,1,1,2,1,1,14,4,1,1,336,60,8,1,1,25836,2960,248,16,1,1,6251504,
%T A132625 454072,24800,1008,32,1,1,4838830976,216266368,7603952,202944,4064,64,
%U A132625 1,1,12344615283200,328381917376,7190266752,124427232,1641856,16320,128,1,1
%N A132625 Triangle T, read by rows, where row n+1 of T = row n of T^(2^n) with appended '1' for n>=0 with T(0,0)=1.
%C A132625 Let R_{n} equal row n of square array A136555, where A136555(n,k) = C(2^k + n-1, k); this triangle transforms rows of A136555: T * R_{n} = R_{n+1} for n>=0.
%e A132625 Triangle begins:
%e A132625 1;
%e A132625 1, 1;
%e A132625 2, 1, 1;
%e A132625 14, 4, 1, 1;
%e A132625 336, 60, 8, 1, 1;
%e A132625 25836, 2960, 248, 16, 1, 1;
%e A132625 6251504, 454072, 24800, 1008, 32, 1, 1;
%e A132625 4838830976, 216266368, 7603952, 202944, 4064, 64, 1, 1;
%e A132625 12344615283200, 328381917376, 7190266752, 124427232, 1641856, 16320, 128, 1, 1; ...
%e A132625 GENERATE T FROM MATRIX POWERS OF T.
%e A132625 Matrix power T^4 begins:
%e A132625 1;
%e A132625 4, 1;
%e A132625 14, 4, 1; <-- row 3 of T
%e A132625 96, 22, 4, 1;
%e A132625 1941, 316, 38, 4, 1;
%e A132625 129206, 14185, 1140, 70, 4, 1; ...
%e A132625 where row 3 of T = row 2 of T^(2^2) with appended '1'.
%e A132625 Matrix power T^8 begins:
%e A132625 1;
%e A132625 8, 1;
%e A132625 44, 8, 1;
%e A132625 336, 60, 8, 1; <-- row 4 of T
%e A132625 6062, 872, 92, 8, 1;
%e A132625 345596, 35734, 2712, 156, 8, 1; ...
%e A132625 where row 4 of T = row 3 of T^(2^3) with appended '1'.
%e A132625 Matrix power T^16 begins:
%e A132625 1;
%e A132625 16, 1;
%e A132625 152, 16, 1;
%e A132625 1504, 184, 16, 1;
%e A132625 25836, 2960, 248, 16, 1; <-- row 5 of T
%e A132625 1197304, 109500, 7408, 376, 16, 1; ...
%e A132625 where row 5 of T = row 4 of T^(2^4) with appended '1'.
%e A132625 Alternate generating method:
%e A132625 RoW 3: start with '1' followed by (2^2 - 1) zeros;
%e A132625 take partial sums and append (2^1 - 1) zero;
%e A132625 take partial sums twice more:
%e A132625 (1), 0, 0, 0;
%e A132625 1, 1, 1, (1), 0;
%e A132625 1, 2, 3, 4, (4);
%e A132625 1, 3, 6, 10, (14);
%e A132625 the final nonzero terms form row 3: [14, 4, 1, 1].
%e A132625 Row 4: start with '1' followed by (2^3 - 1) zeros;
%e A132625 take partial sums and append (2^2 - 1) zeros;
%e A132625 take partial sums and append (2^1 - 1) zero;
%e A132625 take partial sums twice more:
%e A132625 (1), 0, 0, 0, 0, 0, 0, 0;
%e A132625 1, 1, 1, 1, 1, 1, 1, (1), 0, 0, 0;
%e A132625 1, 2, 3, 4, 5, 6, 7, 8, 8, 8, (8), 0;
%e A132625 1, 3, 6, 10, 15, 21, 28, 36, 44, 52, 60, (60);
%e A132625 1, 4, 10, 20, 35, 56, 84, 120, 164, 216, 276, (336);
%e A132625 the final nonzero terms form row 4: [336, 60, 8, 1, 1].
%e A132625 Continuing in this way produces all the rows of this triangle.
%o A132625 (PARI) T(n, k)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(A^(2^(i-2)))[i-1, j]); )); A=B); return( ((A)[n+1, k+1]))
%o A132625 (PARI) /* Generate using partial sums method (faster) */ T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k, p=2^n-2^(n-j)-j; A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A[p+1]
%o A132625 (PARI) /* As Row Transformation of Square Array A136555(n,k) = C(2^k + n-1, k): */ T(n,k)=local(M=matrix(n+2,n+2,r,c,binomial(2^(c-1)+r-2,c-1)), N=matrix(n+1,n+1,r,c,M[r,c]),P=matrix(n+1,n+1,r,c,M[r+1,c]),R=P~*N~^-1); R[n+1,k+1]
%Y A132625 Cf. variants: A101479, A132610, A132615; columns: A132626, A132627.
%Y A132625 Cf. A136555.
%K A132625 nonn,tabl
%O A132625 0,4
%A A132625 _Paul D. Hanna_, Aug 25 2007, Jan 07 2008