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A117467
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The smallest part summed over all partitions of n in which every integer from the smallest part to the largest part occurs.
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3
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MAPLE
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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):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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