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A152406 Triangle T, read by rows, where column k of T = column 0 of matrix power T^(k+1) for k>0, with column 0 of T = column 1 of T^2 (shifted). 4

%I #4 Jun 14 2017 01:12:12

%S 1,1,1,4,2,1,26,10,3,1,224,74,18,4,1,2346,698,150,28,5,1,28516,7838,

%T 1546,260,40,6,1,391042,100850,18642,2916,410,54,7,1,5936376,1451454,

%U 254690,37712,4980,606,70,8,1,98435034,22985130,3861782,547240,68910

%N Triangle T, read by rows, where column k of T = column 0 of matrix power T^(k+1) for k>0, with column 0 of T = column 1 of T^2 (shifted).

%F Column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.

%F Column k: T(n,k) = Sum_{j=0..n-k} T(n-k,j)*T(j+k-1,k-1) for n>=k>0.

%F Column 0: T(n,0) = Sum_{j=1..n} T(n,j)*T(j,1) for n>=0.

%e Triangle T begins:

%e 1;

%e 1, 1;

%e 4, 2, 1;

%e 26, 10, 3, 1;

%e 224, 74, 18, 4, 1;

%e 2346, 698, 150, 28, 5, 1;

%e 28516, 7838, 1546, 260, 40, 6, 1;

%e 391042, 100850, 18642, 2916, 410, 54, 7, 1;

%e 5936376, 1451454, 254690, 37712, 4980, 606, 70, 8, 1;

%e 98435034, 22985130, 3861782, 547240, 68910, 7934, 854, 88, 9, 1;...

%e where column k of T = column 0 of T^(k+1) for k>0

%e and column 0 of T = column 1 of T^2 (shifted).

%e Amazingly, column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.

%e Matrix square T^2 begins:

%e 1;

%e 2, 1;

%e 10, 4, 1;

%e 74, 26, 6, 1;

%e 698, 224, 48, 8, 1;

%e 7838, 2346, 474, 76, 10, 1;

%e 100850, 28516, 5492, 848, 110, 12, 1;

%e 1451454, 391042, 72334, 10804, 1370, 150, 14, 1;...

%e where column 0 of T^2 = column 1 of T,

%e and column 2 of T^2 = column 1 of T^3.

%e Matrix cube T^3 begins:

%e 1;

%e 3, 1;

%e 18, 6, 1;

%e 150, 48, 9, 1;

%e 1546, 474, 90, 12, 1;

%e 18642, 5492, 1032, 144, 15, 1;

%e 254690, 72334, 13362, 1884, 210, 18, 1;

%e 3861782, 1060412, 192192, 27040, 3090, 288, 21, 1;...

%e where column 0 of T^3 = column 2 of T,

%e and column 3 of T^3 = column 2 of T^4.

%e Matrix power T^4 begins:

%e 1;

%e 4, 1;

%e 28, 8, 1;

%e 260, 76, 12, 1;

%e 2916, 848, 144, 16, 1;

%e 37712, 10804, 1884, 232, 20, 1;

%e 547240, 153840, 27040, 3488, 340, 24, 1;

%e 8751688, 2410328, 423240, 55840, 5780, 468, 28, 1;...

%e where column 0 of T^4 = column 3 of T,

%e and column 1 of T^4 = column 3 of T^2.

%o (PARI) T(n, k)=if(k>n || n<0,0, if(k==n,1, if(k==0,sum(j=1,n,T(n,j)*T(j,1)), sum(j=0,n-k,T(n-k, j)*T(j+k-1, k-1)));))

%Y Cf. columns: A152407, A152408, A152409, A152410.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Dec 05 2008

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)