A152790 Triangle T, read by rows, where g.f. of row n of matrix power T^(2^n) = (2^(2n-1) + y)*y^(n-1) for n>0.

1, 1, 1, -3, 2, 1, 84, -28, 4, 1, -12520, 3040, -240, 8, 1, 8233600, -1757824, 103168, -1984, 16, 1, -14411593728, 4551192576, -235382784, 3397632, -16128, 32, 1, -376752260382720, -32793120079872, 2419300630528, -30807851008, 110280704

Offset

0,4

Links

Table of n, a(n) for n=0..32.

Formula

T(n,k) is divisible by 2^(k*(n-k)) for n>=k>=0.

For matrix powers: [T^m](n,k)/2^(k*(n-k)) is an integer for all integer m and n>=k>=0.

GENERATING A152795 FROM MATRIX POWERS of this triangle T:

A152795(n,k) = [T^(2^(n+1))](n-k,0) / 2^((n-k)*(n-k-1)+n+1) for n>k;

A152795(n,k) = [T^(2^(n+m))](n-k+m-1,m-1) / 2^((n-k+m-1)*(n-k+m-2) - (m-1)^2 + n+m) for m>0, n>k, with A152795(n,n)=1.

Example

Triangle T begins:
1;
1,1;
-3,2,1;
84,-28,4,1;
-12520,3040,-240,8,1;
8233600,-1757824,103168,-1984,16,1;
-14411593728,4551192576,-235382784,3397632,-16128,32,1;
-376752260382720,-32793120079872,2419300630528,-30807851008,110280704,-130048,64,1;
2151855694060453888,-2984404225397620736,-70623685744001024,1261902860648448,-3987317719040,3554017280,-1044480,128,1;
...
Illustrate: g.f. of row n of T^(2^n) = (2^(2n-1) + y)*y^(n-1) as follows.
Matrix power T^(2^3) begins:
1;
8,1;
32,16,1;
0,0,32,1; <-- row 3 of T^(2^3), g.f.: (2^5 + y)*y^2
...
Matrix power T^(2^4) begins:
1;
16,1;
192,32,1;
1024,512,64,1;
0,0,0,128,1; <-- row 4 of T^(2^4), g.f.: (2^7 + y)*y^3
...
Matrix power T^(2^5) begins:
1;
32,1;
896,64,1;
22528,3072,128,1;
131072,65536,8192,256,1;
0,0,0,0,512,1; <-- row 5 of T^(2^5), g.f.: (2^9 + y)*y^4
...
Matrix power T^(2^6) begins:
1;
64,1;
3840,128,1;
258048,14336,256,1;
15466496,1441792,49152,512,1;
67108864,33554432,4194304,131072,1024,1;
0,0,0,0,0,2048,1; <-- row 6 of T^(2^6), g.f.: (2^11 + y)*y^5
...
Triangle U resulting from U(n,k) = T(n,k)/2^(k*(n-k)) begins:
1;
1,1;
-3,1,1;
84,-7,1,1;
-12520,380,-15,1,1;
8233600,-109864,1612,-31,1,1;
-14411593728,142224768,-919464,6636,-63,1,1;
...
demonstrating that 2^(k*(n-k)) divides T(n,k).
ILLUSTRATE GENERATION OF TRIANGLE A152795.
Matrix power T^(2^7) begins:
1;
128,1;
15872,256,1;
2416640,61440,512,1;
444071936,16515072,229376,1024,1;
68048388096,3959422976,92274688,786432,2048,1;
137438953472,68719476736,8589934592,268435456,2097152,4096,1;
0,0,0,0,0,0,8192,1; <-- g.f. of row 7 = (2^13 + y)*y^6
...
Now remove all factors of 2 from the first 6 rows to obtain:
1;
1, 1;
31, 1, 1;
295, 15, 1, 1;
847, 63, 7, 1, 1;
507, 59, 11, 3, 1, 1;
1, 1, 1, 1, 1, 1, 1.
Transposing this resultant triangle about the antidiagonal
yields the first 6 rows of triangle A152795:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 11, 7, 1, 1;
1, 59, 63, 15, 1, 1;
1, 507, 847, 295, 31, 1, 1.
This process extracts the initial m-1 rows of triangle A152795 from
the matrix power T^(2^m) for all m>1 -- a very pretty result!

Prog

(PARI) {T(n, k) = my(M, N); M = if(n==0, Mat(1), M=matrix(n, n, r, c, if(r>=c, T(r-1, c-1))));
for(j=1, n, N=matrix(#M+1, #M+1, r, c, if(r==c, 1, if(r>c, if(r<=#M, M[r, c], if(c==#M, 2^(c-1), 0)))));
for(i=2, #N-1, N[ #N, #N-i]=-(N^(2^(#N-1)))[ #N, #N-i]/2^(#N-1)); M=N); M[n+1, k+1]}
for(n=0, 8, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. columns: A152791, A152792, A152793; A152794.

Cf. variants: A152285, A134049, A132870, A132875.

Cf. related triangle: A152795.

Sequence in context: A346743 A087041 A357675 * A247602 A201902 A239893

Adjacent sequences: A152787 A152788 A152789 * A152791 A152792 A152793

Keyword

sign,tabl

Author

Paul D. Hanna, Dec 18 2008, Dec 19 2008

Status

approved