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Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k distinct parts (n>=1, k>=1).
2

%I #12 May 22 2015 05:56:40

%S 1,1,2,1,1,2,1,2,2,2,3,1,5,3,4,1,2,7,1,2,8,2,2,10,3,2,11,5,2,13,7,4,

%T 12,11,1,19,11,1,2,18,17,1,3,20,21,2,2,22,27,3,2,25,32,5,4,24,41,7,2,

%U 30,46,11,2,31,56,15,2,36,62,22,3,33,80,25,1,2,39,87,36,1,4,38,103,45,2,2,45

%N Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k distinct parts (n>=1, k>=1).

%C Row n has floor(sqrt(n)) terms. Row sums yield A000009. T(n,1)=A001227(n) (n>=1). Sum(k*T(n,k),k>=1)=A038348(n-1) (n>=1).

%H Alois P. Heinz, <a href="/A116674/b116674.txt">Rows n = 1..1000, flattened</a>

%F G.f.: product(1+tx^(2j-1)/(1-x^(2j-1)), j=1..infinity).

%e T(9,2)=4 because the only partitions of 9 into odd parts and having 2 distinct parts are [7,1,1],[5,1,1,1,1],[3,3,1,1,1] and [3,1,1,1,1,1,1].

%e Triangle starts:

%e 1;

%e 1;

%e 2;

%e 1,1;

%e 2,1;

%e 2,2;

%e 2,3;

%p g:=product(1+t*x^(2*j-1)/(1-x^(2*j-1)),j=1..35): gser:=simplify(series(g,x=0,34)): for n from 1 to 29 do P[n]:=coeff(gser,x^n) od: for n from 1 to 29 do seq(coeff(P[n],t,j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form

%p # second Maple program:

%p with(numtheory):

%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,

%p `if`(i<1, 0, add(b(n-i*j, i-2)*`if`(j=0, 1, x), j=0..n/i))))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(

%p b(n, iquo(n+1, 2)*2-1)):

%p seq(T(n), n=1..30); # _Alois P. Heinz_, Mar 08 2015

%t b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-2]*If[j == 0, 1, x], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, Quotient[n+1, 2]*2-1]]; Table[T[n], {n, 1, 30}] // Flatten (* _Jean-François Alcover_, May 22 2015, after _Alois P. Heinz_ *)

%Y Cf. A000009, A001227, A038348.

%K nonn,tabf,look

%O 1,3

%A _Emeric Deutsch_, Feb 22 2006