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A125800 Rectangular table where column k equals row sums of matrix power A078122^k, read by antidiagonals. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

LINKS

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

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

EXAMPLE

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.

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-Fran├žois Alcover, Jan 08 2016 *)

PROG

(PARI) T(n, k, p=0, q=3)=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]))

CROSSREFS

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

Sequence in context: A294585 A283674 A294758 * A264698 A263296 A259862

Adjacent sequences:  A125797 A125798 A125799 * A125801 A125802 A125803

KEYWORD

nonn,tabl,look

AUTHOR

Paul D. Hanna, Dec 10 2006

STATUS

approved

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Last modified February 24 18:14 EST 2021. Contains 341577 sequences. (Running on oeis4.)