A276428 - OEIS

%I #23 Feb 01 2024 00:27:41

%S 0,1,0,1,2,3,3,6,7,12,15,22,27,40,49,68,87,116,145,193,239,311,387,

%T 494,611,776,952,1193,1464,1817,2214,2733,3315,4060,4911,5974,7195,

%U 8713,10448,12585,15048,18039,21486,25660,30462,36231,42888,50820,59972,70843,83354

%N Sum over all partitions of n of the number of distinct parts i of multiplicity i.

%H Alois P. Heinz, <a href="/A276428/b276428.txt">Table of n, a(n) for n = 0..1000</a>

%H Philip Cuthbertson, David J. Hemmer, Brian Hopkins, and William J. Keith, <a href="https://arxiv.org/abs/2401.06254">Partitions with fixed points in the sequence of first-column hook lengths</a>, arXiv:2401.06254 [math.CO], 2024.

%F a(n) = Sum_{k>=0} k*A276427(n,k).

%F G.f.: g(x) = Sum_{i>=1} (x^{i^2}*(1-x^i))/Product_{i>=1} (1-x^i).

%e 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.

%p 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);

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],

%p      `if`(i<1, 0, add((p-> p+`if`(i<>j, 0,

%p       [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 19 2016

%t 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 *)

%o (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

%Y Cf. A276427, A276434, A277101; A114638, A116861.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Sep 19 2016