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A116856 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts such that the smallest part is k (n>=1, k>=1).

%I A116856 #8 Aug 26 2024 11:43:00
%S A116856 1,1,1,0,1,2,2,0,0,0,1,3,0,1,4,0,0,0,0,0,1,5,0,1,6,0,1,0,0,0,0,0,1,8,
%T A116856 0,1,0,1,10,0,1,0,0,0,0,0,0,0,1,12,0,2,0,1,15,0,2,0,0,0,0,0,0,0,0,0,1,
%U A116856 18,0,2,0,1,0,1,22,0,3,0,1,0,0,0,0,0,0,0,0,0,1,27,0,3,0,1,0,1,32,0,4,0,1,0
%N A116856 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts such that the smallest part is k (n>=1, k>=1).
%C A116856 Row 2n-1 has 2n-1 terms; row 4n+2 has 2n+1 terms; row 4n has 2n-1 terms. Row sums yield A000009.
%F A116856 G.f.: Sum_{j=1..oo} t^(2*j-1)*x^(2*j-1)/Product_{i=j..oo} 1-x^(2*i-1).
%F A116856 T(n,2k)=0.
%F A116856 Sum_{k>=1} k*T(n,k) = A092314(n).
%e A116856 T(12,3)=2 because we have [9,3] and [3,3,3,3].
%e A116856 Triangle starts:
%e A116856 1;
%e A116856 1;
%e A116856 1,0,1;
%e A116856 2;
%e A116856 2,0,0,0,1;
%e A116856 3,0,1;
%e A116856 4,0,0,0,0,0,1
%p A116856 g:=sum(t^(2*j-1)*x^(2*j-1)/product(1-x^(2*i-1),i=j..20),j=1..30): gser:=simplify(series(g,x=0,20)): for n from 1 to 17 do P[n]:=sort(coeff(gser,x^n)) od: d:=proc(n) if n mod 2 = 1 then n elif n mod 4 = 2 then n/2 else n/2-1 fi end: for n from 1 to 17 do seq(coeff(P[n],t^j),j=1..d(n)) od; # yields sequence in triangular form; d(n) is the degree of the polynomial P[n]
%t A116856 imax = 18;
%t A116856 Rest@CoefficientList[#, t]& /@ Rest@CoefficientList[Sum[t^(2j-1)*x^(2j-1)/ Product[1 - x^(2i-1), {i, j, imax}], {j, 1, imax}] + O[x]^imax, x] // Flatten (* _Jean-François Alcover_, Aug 26 2024 *)
%Y A116856 Cf. A000009, A092314, A116799.
%K A116856 nonn,tabf
%O A116856 1,6
%A A116856 _Emeric Deutsch_, Feb 24 2006