Robert Israel, <a href="/A125800/b125800_1.txt">Table of n, a(n) for n = 0..2484</a> (antidiagonals 0 to 69, flattened)

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OEIS Server: Installed new b-file as b125800.txt.  Old b-file is now b125800_1.txt.

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

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

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.

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.

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, ...;

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; ...

1;

2, , 1;

5, , 6, , 1;

23, , 51, , 18, , 1;

239, , 861, , 477, , 54, , 1;

5828, 32856, 25263, 4347, 162, 1; ...

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Jon E. Schoenfield: I don't understand why the Example section repeatedly refers to "this table A125790" (and once, self-referentially, to "this table A125800).

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T:= proc(n, k) option remember;

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:

if kseq(seq(coeff(S[s-i], z, i), i=0 or n..s), s=0..20); # _Robert Israel_, Jun then02 12019

else procname(n, k-1)+procname(n-1, 3*k)

fi

end proc:

seq(seq(T(s-i, i), i=0..s), s=0..10); # Robert Israel, Jun 02 2019

Robert Israel, <a href="/A125800/b125800_1.txt">Table of n, a(n) for n = 0..1892484</a>(> (antidiagonals 0 to 1869, flattened)

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

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nonn,tabl,changed,look