A117467 The smallest part summed over all partitions of n in which every integer from the smallest part to the largest part occurs.

1, 3, 5, 8, 10, 15, 17, 22, 28, 32, 37, 49, 52, 60, 77, 83, 94, 116, 125, 146, 174, 187, 214, 257, 282, 315, 372, 410, 461, 544, 593, 669, 773, 851, 969, 1105, 1218, 1368, 1559, 1737, 1936, 2199, 2431, 2717, 3079, 3396, 3790, 4263, 4719, 5262, 5878, 6501, 7224

Offset: 1

Author

Emeric Deutsch, Mar 19 2006

Keywords

• nonn

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..5000 (terms 1..1000 from Alois P. Heinz)

Crossrefs

Cf. A117466.

Programs

Maple

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);
# second Maple program:
b:= proc(n, k, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(in, 0, b(n-i, k, i)))))
    end:
T:= (n, k)-> add(b(n-(i+k)*(i+1-k)/2, k, i), i=k..n):
a:= n-> add(T(n, k)*k, k=1..n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 04 2015

Mathematica

b[n_, k_, i_] := b[n, k, i] = If[n<0, 0, If[n == 0, 1, If[in, 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 *)
nmax = 100; p = 1; s = 1; Do[p *= x*(1 + x^k); s += p/(1 - x^k)^2/(1 + x^k);, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 14 2025 *)

Formula

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).

a(n) = Sum_{k=1..n} k * A117466(n,k).

a(n) ~ exp(Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 16 2025