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A263232 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 3 (n >= 0, 0 <= k <= floor(n/3)).

%I A263232 #16 May 25 2019 01:54:08
%S A263232 1,1,2,2,1,4,1,5,2,8,2,1,10,4,1,15,5,2,19,8,2,1,27,10,4,1,34,15,5,2,
%T A263232 47,19,8,2,1,59,27,10,4,1,79,34,15,5,2,99,47,19,8,2,1,130,59,27,10,4,
%U A263232 1,162,79,34,15,5,2,209,99,47,19,8,2,1,259,130,59,27,10,4,1
%N A263232 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 3 (n >= 0, 0 <= k <= floor(n/3)).
%C A263232 Row n has 1+floor(n/3) terms. Row sums are the partition numbers (A000041). T(n,0) = A027337(n). Sum_{k=0..floor(n/3)} k*T(n,k) = A024787(n).
%H A263232 Alois P. Heinz, <a href="/A263232/b263232.txt">Rows n = 0..300, flattened</a>
%F A263232 G.f.: (1-x)*(1-x^2)*(1-tx^3)*Product_{j>=4} (1-x^j).
%e A263232 T(7,1) = 4 because we have [4,3], [3,2,2], [3,2,1,1], and [3,1,1,1,1].
%e A263232 T(9,2) = 2 because we have [3,3,2,1] and [3,3,1,1,1].
%e A263232 Triangle starts:
%e A263232   1;
%e A263232   1;
%e A263232   2;
%e A263232   2, 1;
%e A263232   4, 1;
%e A263232   5, 2;
%e A263232   8, 2, 1.
%p A263232 g := 1/((1-x)*(1-x^2)*(1-t*x^3)*(product(1-x^j, j = 4 .. 80))): gser := simplify(series(g, x = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
%p A263232 # second Maple program:
%p A263232 b:= proc(n, i) option remember; expand(
%p A263232       `if`(n=0, 1, `if`(i<1, 0, `if`(i=3, x, 1)*
%p A263232       `if`(i>n, 0, b(n-i, i)) +b(n, i-1))))
%p A263232     end:
%p A263232 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A263232 seq(T(n), n=0..25);  # _Alois P. Heinz_, Nov 01 2015
%t A263232 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, If[i == 3, x, 1]* If[i > n, 0, b[n - i, i]] + b[n, i - 1]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 25}] // Flatten (* _Jean-François Alcover_, Jan 21 2016, after _Alois P. Heinz_ *)
%Y A263232 Cf. A000041, A027337, A024787, A116599.
%K A263232 nonn,tabf
%O A263232 0,3
%A A263232 _Emeric Deutsch_, Nov 01 2015