A138271 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^4 shift right.
1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 268, 78, 18, 4, 1, 3164, 798, 156, 28, 5, 1, 43672, 9874, 1714, 268, 40, 6, 1, 682632, 141282, 22368, 3164, 420, 54, 7, 1, 11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1, 224283416, 40400466, 5566728, 682632, 77720
Offset: 0
Examples
Triangle T begins: 1; 1, 1; 4, 2, 1; 28, 10, 3, 1; 268, 78, 18, 4, 1; 3164, 798, 156, 28, 5, 1; 43672, 9874, 1714, 268, 40, 6, 1; 682632, 141282, 22368, 3164, 420, 54, 7, 1; 11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1; 224283416, 40400466, 5566728, 682632, 77720, 8378, 868, 88, 9, 1; ... where column k of T = column 0 of T^(k+1) with column 0 of T = column 0 of T^4 shift right: column 1 of T = column 0 of T^2; column 2 of T = column 0 of T^3; column 3 of T = column 0 of T^4. Matrix square of T, T^2, begins: 1; 2, 1; 10, 4, 1; 78, 26, 6, 1; 798, 232, 48, 8, 1; 9874, 2578, 486, 76, 10, 1; 141282, 33764, 5888, 864, 110, 12, 1; 2273730, 503910, 82210, 11396, 1390, 150, 14, 1; ... where column k of T^2 = column 1 of T^(k+1): column 0 of T^2 = column 1 of T; column 2 of T^2 = column 1 of T^3; column 3 of T^2 = column 1 of T^4. Matrix cube of T, T^3, begins: 1; 3, 1; 18, 6, 1; 156, 48, 9, 1; 1714, 486, 90, 12, 1; 22368, 5888, 1050, 144, 15, 1; 333910, 82210, 14046, 1908, 210, 18, 1; 5566728, 1289928, 211182, 28072, 3120, 288, 21, 1; ... where column k of T^3 = column 2 of T^(k+1): column 0 of T^3 = column 2 of T; column 1 of T^3 = column 2 of T^2; column 3 of T^3 = column 2 of T^4. Matrix 4th power of T, T^4, begins: 1; 4, 1; 28, 8, 1; 268, 76, 12, 1; 3164, 864, 144, 16, 1; 43672, 11396, 1908, 232, 20, 1; 682632, 170000, 28072, 3520, 340, 24, 1; 11834536, 2814832, 454848, 57408, 5820, 468, 28, 1; ... where column k of T^4 = column 3 of T^(k+1): column 0 of T^4 = column 3 of T = column 0 of T shift left; column 1 of T^4 = column 3 of T^2; column 2 of T^4 = column 3 of T^3.
Crossrefs
Programs
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PARI
{T(n, k) = if(k>n||k<0, 0, if(k==n, 1, if(k==0, T(n+2, 3), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )} for(n=0,10,for(k=0,n,print1(T(n,k),", "));print("")) -
PARI
/* Column k of T = column 0 of T^(k+1): */ {T(n, k) = local(M=if(n==0,Mat(1),matrix(n,n,r,c,if(r>=c,T(r-1,c-1))))); if(k==n, 1, if(k==0, (M^4)[n, 1],(M^(k+1))[n-k+1, 1]))} for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
Formula
Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.