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 A276428 Sum over all partitions of n of the number of distinct parts i of multiplicity i. 10
 0, 1, 0, 1, 2, 3, 3, 6, 7, 12, 15, 22, 27, 40, 49, 68, 87, 116, 145, 193, 239, 311, 387, 494, 611, 776, 952, 1193, 1464, 1817, 2214, 2733, 3315, 4060, 4911, 5974, 7195, 8713, 10448, 12585, 15048, 18039, 21486, 25660, 30462, 36231, 42888, 50820, 59972, 70843, 83354 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{k>=0} k*A276427(n,k). G.f.: g(x) = Sum_{i>=1} (x^{i^2}*(1-x^i))/Product_{i>=1} (1-x^i). EXAMPLE a(5) = 3 because in the partitions [1,1,1,1,1], [1,1,1,2], [1',2',2], [1,1,3], [2,3], [1',4], [5] of 5 only the marked parts satisfy the requirement. MAPLE g := (sum(x^(i^2)*(1-x^i), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50); # second Maple program: b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add((p-> p+`if`(i<>j, 0, [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i))) end: a:= n-> b(n\$2)[2]: seq(a(n), n=0..60); # Alois P. Heinz, Sep 19 2016 MATHEMATICA b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i==j, x, 1]*b[n - i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; a[n_] := (row = T[n]; row.Range[0, Length[row]-1]); Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, Nov 28 2016, after Alois P. Heinz's Maple code for A276427 *) PROG (PARI) apply( A276428(n, s, c)={forpart(p=n, c=1; for(i=1, #p, p[i]==if(i<#p, p[i+1])&&c++&&next; c==p[i]&&s++; c=1)); s}, [0..20]) \\ M. F. Hasler, Oct 27 2019 CROSSREFS Cf. A276427, A276434, A277101; A114638, A116861. Sequence in context: A121833 A091606 A027037 * A020878 A158278 A187505 Adjacent sequences: A276425 A276426 A276427 * A276429 A276430 A276431 KEYWORD nonn AUTHOR Emeric Deutsch, Sep 19 2016 STATUS approved

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Last modified March 31 07:39 EDT 2023. Contains 361645 sequences. (Running on oeis4.)