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%I #11 Jun 29 2015 05:06:22
%S 1,3,5,8,10,15,17,22,28,32,37,49,52,60,77,83,94,116,125,146,174,187,
%T 214,257,282,315,372,410,461,544,593,669,773,851,969,1105,1218,1368,
%U 1559,1737,1936,2199,2431,2717,3079,3396,3790,4263,4719,5262,5878,6501,7224
%N The smallest part summed over all partitions of n in which every integer from the smallest part to the largest part occurs.
%H Alois P. Heinz, <a href="/A117467/b117467.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: sum(x^j*product(1+x^i, i=1..j-1)/(1-x^j)^2, j=1..infinity) (obtained by taking the derivative with respect to t of the g.f. G(t,x) of A117466 and setting t=1).
%F a(n) = Sum_{k=1..n} k * A117466(n,k).
%e a(5)=10 because in the 5 (=A034296(5)) partitions in which every integer from the smallest to the largest part occurs, namely [5],[3,2],[2,2,1],[2,1,1,1] and [1,1,1,1,1], the sum of the smallest parts is 5+2+1+1+1=10.
%p g:=sum(x^j*product(1+x^i,i=1..j-1)/(1-x^j)^2,j=1..70): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=1..60);
%p # second Maple program:
%p b:= proc(n, k, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p `if`(i<k, 0, b(n, k, i-1)+`if`(i>n, 0, b(n-i, k, i)))))
%p end:
%p T:= (n, k)-> add(b(n-(i+k)*(i+1-k)/2, k, i), i=k..n):
%p a:= n-> add(T(n, k)*k, k=1..n):
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Jun 04 2015
%t b[n_, k_, i_] := b[n, k, i] = If[n<0, 0, If[n == 0, 1, If[i<k, 0, b[n, k, i-1] + If[i>n, 0, b[n-i, k, i]]]]]; T[n_, k_] := Sum[b[n-(i+k)*(i+1-k)/2, k, i], {i, k, n}]; a[n_] := Sum[T[n, k]*k, {k, 1, n}]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Jun 29 2015, after _Alois P. Heinz_ *)
%Y Cf. A117466.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Mar 19 2006
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