A125800 Rectangular table where column k equals row sums of matrix power A078122^k, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 23, 12, 4, 1, 1, 239, 93, 22, 5, 1, 1, 5828, 1632, 238, 35, 6, 1, 1, 342383, 68457, 5827, 485, 51, 7, 1, 1, 50110484, 7112055, 342382, 15200, 861, 70, 8, 1, 1, 18757984046, 1879090014, 50110483, 1144664, 32856, 1393, 92, 9, 1

Offset: 0

Author

Paul D. Hanna, Dec 10 2006

Keywords

• nonn
• tabl
• look

Comments

Determinant of n X n upper left submatrix is 3^(n*(n-1)*(n-2)/6).

This table is related to partitions of numbers into powers of 3 (see A078122).

Triangle A078122 shifts left one column under matrix cube.

Column 1 is A078125, which equals row sums of A078122;

column 2 is A078124, which equals row sums of A078122^2.

Examples

			Recurrence T(n,k) = T(n,k-1) + T(n-1,3*k) is illustrated by:
  T(3,3) = T(3,2) + T(2,9) = 93 + 145 = 238;
  T(4,3) = T(4,2) + T(3,9) = 1632 + 4195 = 5827;
  T(5,3) = T(5,2) + T(4,9) = 68457 + 273925 = 342382.
Rows of this table begin:
  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, ...;
  1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, ...;
  1, 23, 93, 238, 485, 861, 1393, 2108, 3033, 4195, 5621, ...;
  1, 239, 1632, 5827, 15200, 32856, 62629, 109082, 177507, 273925,...;
  1, 5828, 68457, 342382, 1144664, 3013980, 6769672, 13570796, ...;
  1, 342383, 7112055, 50110483, 215155493, 690729981, 1828979530, ...;
  1, 50110484, 1879090014, 18757984045, 103674882878, 406279238154,..;
  1, 18757984046, 1287814075131, 18318289003447, 130648799730635, ...;
Triangle A078122 begins:
  1;
  1,     1;
  1,     3,      1;
  1,    12,      9,     1;
  1,    93,    117,    27,    1;
  1,  1632,   3033,  1080,   81,   1;
  1, 68457, 177507, 86373, 9801, 243, 1; ...
where row sums form column 1 of this table A125790,
and column k of A078122 equals column 3^k - 1 of this table A125800.
Matrix square A078122^2 begins:
     1;
     2,     1;
     5,     6,     1;
    23,    51,    18,    1;
   239,   861,   477,   54,   1;
  5828, 32856, 25263, 4347, 162, 1; ...
where row sums form column 2 of this table A125790,
and column 0 of A078122^2 forms column 1 of this table A125790.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2484 (antidiagonals 0 to 69, flattened)

Crossrefs

Cf. A078122; columns: A078125, A078124, A125801, A125802, A125803; A125804 (diagonal), A125805 (antidiagonal sums); related table: A125800 (q=2).

Programs

Maple

f[0]:= 1/(1-z):
S[0]:= series(f[0],z,21):
for n from 1 to 20 do
  ff:= unapply(f[n-1],z);
  f[n]:= simplify(1/3*sum(ff(w*z^(1/3)),w=RootOf(Z^3-1,Z)))/(1-z);
  S[n]:= series(f[n],z,21-n)
od:
seq(seq(coeff(S[s-i],z,i),i=0..s),s=0..20); # Robert Israel, Jun 02 2019

Mathematica

T[0, ] = T[, 0] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 3 k]; Table[T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)

PARI

T(n,k,p=0,q=3)=local(A=Mat(1), B); if(n

				

Formula

T(n,k) = T(n,k-1) + T(n-1,3*k) for n > 0, k > 0, with T(0,n)=T(n,0)=1 for n >= 0.

G.f. of row n is g_n(z) where g_{n+1}(z) = (1-z)^(-1)*Sum_{w^3=1} g_n(w*z^(1/3)) (the sum being over the cube roots of unity). - Robert Israel, Jun 02 2019