A141712 Triangle T, read by rows, where the n-th diagonal of T equals the BINOMIAL transform of the (n-1)-th diagonal of T^2 for n>=1, with the zeroth diagonal set to all 1's and where T^2 denotes the matrix square of T.

1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 26, 26, 18, 8, 1, 162, 162, 114, 54, 16, 1, 1454, 1454, 1030, 506, 162, 32, 1, 18854, 18854, 13394, 6666, 2274, 486, 64, 1, 354258, 354258, 251962, 126134, 43798, 10346, 1458, 128, 1, 9671546, 9671546, 6882102, 3453110, 1210226

Offset

0,4

Links

Table of n, a(n) for n=0..49.

Formula

T(n,k) = Sum_{j=0..k} C(k,j)*[T^2](n-k+j-1,j) for n>k>=0, with T(n,n)=1, for n>=0.

Example

Triangle T begins:
1;
1, 1;
2, 2, 1;
6, 6, 4, 1;
26, 26, 18, 8, 1;
162, 162, 114, 54, 16, 1;
1454, 1454, 1030, 506, 162, 32, 1;
18854, 18854, 13394, 6666, 2274, 486, 64, 1;
354258, 354258, 251962, 126134, 43798, 10346, 1458, 128, 1; ...
Matrix square T^2 begins:
1;
2, 1;
6, 4, 1;
26, 20, 8, 1;
162, 136, 68, 16, 1;
1454, 1292, 732, 236, 32, 1;
18854, 17400, 10648, 4036, 836, 64, 1; ...
where the BINOMIAL transform of diagonal 2 of T^2:
BINOMIAL[6,20,68,236,836,3020,11108,41516,...]
equals: [6,26,114,506,2274,10346,47634,221786,...]
which is diagonal 3 of T.
Specific examples:
T(4,1) = [T^2](2,0) + [T^2](3,1) = 6 + 20 = 26;
T(4,2) = [T^2](1,0) + 2*[T^2](2,1) + [T^2](3,2) = 2 + 2*4 + 8 = 18;
T(5,2) = [T^2](2,0) + 2*[T^2](3,1) + [T^2](4,2) = 6 + 2*20 + 68 = 114;
T(5,3) = [T^2](1,0) + 3*[T^2](2,1) + 3*[T^2](3,2) + [T^2](4,3) = 2 + 3*4 + 3*8 + 16 = 54.

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, if(n==k+1, 2^k, if(k==1, T(n, 0), sum(j=0, k, binomial(k, j)*sum(i=0, n-k+j-1, T(n-k+j-1, i)*T(i, j)))))))

Crossrefs

Cf. A141713 (column 0), A141714 (column 2); A141715 (T^2), A141716.

Sequence in context: A225112 A108076 A104557 * A098539 A222073 A343847

Adjacent sequences: A141709 A141710 A141711 * A141713 A141714 A141715

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 01 2008

Status

approved