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

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

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).

Example

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.

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`(i<k, 0, b(n, k, i-1)+`if`(i>n, 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[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 *)

Crossrefs

Cf. A117466.

Sequence in context: A187973 A190488 A182082 * A310025 A133097 A117176

Adjacent sequences: A117464 A117465 A117466 * A117468 A117469 A117470

Keyword

nonn

Author

Emeric Deutsch, Mar 19 2006

Status

approved