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

Author

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

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

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 *)

PARI

{T(n,k,p=0,q=2)=local(A=Mat(1), B); if(n
				

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.

Conjecture: g.f. for n-th row is (Sum_{i=0..n-1} x^i Sum_{j=0..i} binomial(n+1,j)*T(n,i-j)*(-1)^j)/(1-x)^(n+1) for n > 0. - Mikhail Kurkov, May 03 2025