A125790 Rectangular table where column k equals row sums of matrix power A078121^k, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 36, 35, 16, 5, 1, 1, 202, 201, 84, 25, 6, 1, 1, 1828, 1827, 656, 165, 36, 7, 1, 1, 27338, 27337, 8148, 1625, 286, 49, 8, 1, 1, 692004, 692003, 167568, 25509, 3396, 455, 64, 9, 1, 1, 30251722, 30251721, 5866452, 664665, 64350, 6321, 680, 81, 10, 1

Offset

0,5

Comments

Determinant of n X n upper left submatrix is 2^[n(n-1)(n-2)/6] (see A125791). Related to partitions of numbers into powers of 2 (see A078121). Triangle A078121 shifts left one column under matrix square.

Links

Table of n, a(n) for n=0..65.

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 5.

Formula

T(n,k) = T(n,k-1) + T(n-1,2*k) for n>0, k>0, with T(0,n)=T(n,0)=1 for n>=0.

Example

Recurrence T(n,k) = T(n,k-1) + T(n-1,2*k) is illustrated by:
T(4,3) = T(4,2) + T(3,6) = 201 + 455 = 656;
T(5,3) = T(5,2) + T(4,6) = 1827 + 6321 = 8148;
T(6,3) = T(6,2) + T(5,6) = 27337 + 140231 = 167568.
Rows of this table begin:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...;
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, ...;
1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, ...;
1, 36, 201, 656, 1625, 3396, 6321, 10816, 17361, 26500, 38841, ...;
1, 202, 1827, 8148, 25509, 64350, 140231, 274856, 497097, ...;
1, 1828, 27337, 167568, 664665, 2026564, 5174449, 11622976, ...;
1, 27338, 692003, 5866452, 29559717, 109082974, 326603719, ...;
1, 692004, 30251721, 356855440, 2290267225, 10243585092, ...; ...
Triangle A078121 begins:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 10, 16, 8, 1;
1, 36, 84, 64, 16, 1;
1, 202, 656, 680, 256, 32, 1; ...
where row sums form column 1 of this table A125790,
and column k of A078121 equals column 2^k-1 of this table A125790.
Matrix cube A078121^3 begins:
1;
3, 1;
9, 6, 1;
35, 36, 12, 1;
201, 286, 144, 24, 1;
1827, 3396, 2300, 576, 48, 1; ...
where row sums form column 3 of this table A125790,
and column 0 of A078121^3 forms column 2 of this table A125790.

Mathematica

T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 2*k]; T[0, _] = T[_, 0] = 1; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 15 2015 *)

Prog

(PARI) {T(n, k, p=0, q=2)=local(A=Mat(1), B); if(n<p||p<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i||j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return((A^(k+1))[n+1, p+1]))}
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A078121; A002577; A125791; columns: A002577, A125792, A125793, A125794, A125795, A125796; diagonals: A125797, A125798; A125799 (antidiagonal sums); related table: A125800 (q=3).

Sequence in context: A112705 A070895 A127054 * A370005 A294082 A129705

Adjacent sequences: A125787 A125788 A125789 * A125791 A125792 A125793

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 10 2006, corrected Dec 12 2006

Status

approved