cp's OEIS Frontend

A264404 Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity greater than 1 is k (0<=k). For example, in the partition [3,2,2,1,1,1] the sum k is 2 + 1 = 3.

%I A264404 #23 Dec 22 2016 11:07:08
%S A264404 1,1,1,1,2,1,2,2,1,3,3,1,4,4,1,2,5,6,2,2,6,8,3,3,2,8,11,4,5,2,10,14,5,
%T A264404 7,3,3,12,19,7,10,5,3,15,24,9,15,6,4,4,18,31,12,20,9,7,4,22,39,15,26,
%U A264404 13,9,6,5,27,49,19,36,17,13,10,5,32,61,24,46,23,18,14,7,6,38,76,30,60,31,24
%N A264404 Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity greater than 1 is k (0<=k). For example, in the partition [3,2,2,1,1,1] the sum k is 2 + 1 = 3.
%C A264404 Only one copy of each part of multiplicity greater than one is used.
%C A264404 Row n contains floor(n/2) entries (n>=0).
%C A264404 Row sums yield the partition numbers (A000041).
%C A264404 T(n,0) = A000009(n).
%C A264404 Sum_{k>=0} k*T(n,k) = A103650(n).
%H A264404 Alois P. Heinz, <a href="/A264404/b264404.txt">Rows n = 0..350, flattened</a>
%F A264404 G.f.: G(t,z) = Product_{j>=1} (1 + x^j + t^j*x^{2*j}/(1 - x^j)).
%e A264404 T(9,3) = 5 because we have [3,3,3], [3,3,2,1], [3,2,2,1,1], [2,2,2,1,1,1], and [2,2,1,1,1,1,1].
%e A264404 Triangle starts:
%e A264404 1;
%e A264404 1;
%e A264404 1,1;
%e A264404 2,1;
%e A264404 2,2,1;
%e A264404 3,3,1;
%e A264404 4,4,1,2;
%p A264404 g := product(1+x^j+t^j*x^(2*j)/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 35)): 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, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%p A264404 # second Maple program:
%p A264404 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A264404       add(expand(b(n-i*j, i-1)*x^`if`(j>1, i, 0)), j=0..n/i)))
%p A264404     end:
%p A264404 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A264404 seq(T(n), n=0..20);  # _Alois P. Heinz_, Nov 29 2015
%t A264404 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Expand[b[n - i*j, i - 1]*x^If[j > 1, i, 0]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Dec 22 2016, after _Alois P. Heinz_ *)
%Y A264404 Cf. A000009, A000041, A103650.
%K A264404 nonn,tabf
%O A264404 0,5
%A A264404 _Emeric Deutsch_, Nov 27 2015