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%I #25 Jun 03 2019 00:01:21
%S 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,
%T 1,1,342383,68457,5827,485,51,7,1,1,50110484,7112055,342382,15200,861,
%U 70,8,1,1,18757984046,1879090014,50110483,1144664,32856,1393,92,9,1
%N Rectangular table where column k equals row sums of matrix power A078122^k, read by antidiagonals.
%C Determinant of n X n upper left submatrix is 3^(n*(n-1)*(n-2)/6).
%C This table is related to partitions of numbers into powers of 3 (see A078122).
%C Triangle A078122 shifts left one column under matrix cube.
%C Column 1 is A078125, which equals row sums of A078122;
%C column 2 is A078124, which equals row sums of A078122^2.
%H Robert Israel, <a href="/A125800/b125800.txt">Table of n, a(n) for n = 0..2484</a> (antidiagonals 0 to 69, flattened)
%F 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.
%F 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
%e Recurrence T(n,k) = T(n,k-1) + T(n-1,3*k) is illustrated by:
%e T(3,3) = T(3,2) + T(2,9) = 93 + 145 = 238;
%e T(4,3) = T(4,2) + T(3,9) = 1632 + 4195 = 5827;
%e T(5,3) = T(5,2) + T(4,9) = 68457 + 273925 = 342382.
%e Rows of this table begin:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...;
%e 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, ...;
%e 1, 23, 93, 238, 485, 861, 1393, 2108, 3033, 4195, 5621, ...;
%e 1, 239, 1632, 5827, 15200, 32856, 62629, 109082, 177507, 273925,...;
%e 1, 5828, 68457, 342382, 1144664, 3013980, 6769672, 13570796, ...;
%e 1, 342383, 7112055, 50110483, 215155493, 690729981, 1828979530, ...;
%e 1, 50110484, 1879090014, 18757984045, 103674882878, 406279238154,..;
%e 1, 18757984046, 1287814075131, 18318289003447, 130648799730635, ...;
%e Triangle A078122 begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 12, 9, 1;
%e 1, 93, 117, 27, 1;
%e 1, 1632, 3033, 1080, 81, 1;
%e 1, 68457, 177507, 86373, 9801, 243, 1; ...
%e where row sums form column 1 of this table A125790,
%e and column k of A078122 equals column 3^k - 1 of this table A125800.
%e Matrix square A078122^2 begins:
%e 1;
%e 2, 1;
%e 5, 6, 1;
%e 23, 51, 18, 1;
%e 239, 861, 477, 54, 1;
%e 5828, 32856, 25263, 4347, 162, 1; ...
%e where row sums form column 2 of this table A125790,
%e and column 0 of A078122^2 forms column 1 of this table A125790.
%p f[0]:= 1/(1-z):
%p S[0]:= series(f[0],z,21):
%p for n from 1 to 20 do
%p ff:= unapply(f[n-1],z);
%p f[n]:= simplify(1/3*sum(ff(w*z^(1/3)),w=RootOf(Z^3-1,Z)))/(1-z);
%p S[n]:= series(f[n],z,21-n)
%p od:
%p seq(seq(coeff(S[s-i],z,i),i=0..s),s=0..20); # _Robert Israel_, Jun 02 2019
%t 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 *)
%o (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]))
%Y Cf. A078122; columns: A078125, A078124, A125801, A125802, A125803; A125804 (diagonal), A125805 (antidiagonal sums); related table: A125800 (q=2).
%K nonn,tabl,look
%O 0,5
%A _Paul D. Hanna_, Dec 10 2006
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