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%I #9 Nov 23 2018 03:18:33
%S 1,0,0,1,2,0,1,1,0,2,3,0,2,1,1,2,4,0,0,3,3,1,2,2,5,0,1,3,4,1,0,4,3,2,
%T 6,0,2,3,5,1,1,4,4,2,7,0,0,5,3,3,6,1,2,4,5,2,8,0,1,5,4,3,7,1,0,6,3,4,
%U 6,2,9,0,2,5,5,3,8,1,1,6,4,4,7,2,10,0
%N Irregular triangle T with the nontrivial solutions of 2*e2 + 3*e3 = n, for n >= 2, with nonnegative e2 and e3, ordered as pairs with increasing e2 values.
%C The length of row n is 2*A(n), with A(n) = A008615(n+2) for n >= 2: 2*[1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, ...].
%C The trivial solution for n = 0 is [0, 0]. There is no solution for n = 1.
%C The row sums are given in A321202.
%C If a partition of n with parts 2 or 3 (with inclusive or) is written as 2^{e2} 3^{e3}, where e2 and e3 are nonnegative numbers, then in row n, all pairs [e2, e3] are given, for n >= 2, ordered with increasing values of e2.
%C The corresponding irregular triangle with the multinomial numbers n!/((n - (e2 + e3)!*e2!*e3!) is given in A321203. It gives the coefficients of x^n = x^{2*{e2} + 3*{e3}} of (1 + x^2 + x^3)^n, for n >= 2.
%F T(n, k) gives all pairs [e2, e3] solving 2*e2 + 3*e3 = n, ordered with increasing value of e2, for n >= 2. The trivial solution [0, 0] for n = 0 is not recorded. There is no solution for n = 1.
%e The triangle T(n, k) begins (pairs are separated by commas):
%e n\k 0 1 2 3 4 5 6 7 ...
%e 2: 1 0
%e 3: 0 1
%e 4: 2 0
%e 5: 1 1
%e 6: 0 2, 3 0
%e 7: 2 1
%e 8: 1 2, 4 0
%e 9: 0 3, 3 1
%e 10: 2 2, 5 0
%e 11: 1 3, 4 1
%e 12: 0 4, 3 2, 6 0
%e 13: 2 3, 5 1,
%e 14: 1 4, 4 2, 7 0
%e 15: 0 5, 3 3, 6 1
%e 16: 2 4, 5 2, 8 0
%e 17: 1 5, 4 3, 7 1
%e 18: 0 6, 3 4, 6 2, 9 0
%e 19: 2 5, 5 3, 8 1
%e 20: 1 6, 4 4, 7 2, 10 0
%e ...
%e n=8: the two solutions of 2*e2 + 3*e3 = 8 are [e2, e3] = [1, 2] and = [4, 0], and 1 < 4, therefore row 8 is 1 2 4 0, with a comma after the first pair.
%t row[n_] := Reap[Do[If[2 e2 + 3 e3 == n, Sow[{e2, e3}]], {e2, 0, n/2}, {e3, 0, n/3}]][[2, 1]];
%t Table[row[n], {n, 2, 20}] // Flatten (* _Jean-François Alcover_, Nov 23 2018 *)
%Y Cf. A008615, A321202, A321203.
%K nonn,tabf
%O 2,5
%A _Wolfdieter Lang_, Nov 05 2018
%E Missing row 2 inserted by _Jean-François Alcover_, Nov 23 2018
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