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A277102 Number of partitions of n containing no part i of multiplicity i-1.

%I A277102 #15 Jun 12 2025 11:15:16
%S A277102 1,1,1,2,4,5,7,10,15,21,28,37,50,67,88,115,150,193,248,317,402,508,
%T A277102 640,802,1002,1248,1545,1908,2351,2887,3532,4313,5251,6377,7724,9334,
%U A277102 11254,13541,16253,19473,23286,27791,33100,39362,46723,55370,65504,77377,91257,107477,126380
%N A277102 Number of partitions of n containing no part i of multiplicity i-1.
%H A277102 Vaclav Kotesovec, <a href="/A277102/b277102.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..1000 from Alois P. Heinz)
%F A277102 a(n) = A277100(n,0).
%F A277102 G.f.: g(x) = Product_{i>=1}(1/(1-x^(i+1)) - x^(i(i+1))).
%e A277102 a(4) = 4 because we have [1,1,1,1], [1,3], [2,2], and [4]; the partition [1,1,2] does not qualify.
%p A277102 g := (product(1/(1-x^(i+1))-x^(i*(i+1)), i = 1 .. 100))/(1-x): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
%p A277102 # second Maple program:
%p A277102 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A277102       add(`if`(i-1=j, 0, b(n-i*j, i-1)), j=0..n/i)))
%p A277102     end:
%p A277102 a:= n-> b(n$2):
%p A277102 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 10 2016
%t A277102 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[If[i-1 == j, 0, b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Dec 11 2016 after _Alois P. Heinz_ *)
%Y A277102 Cf. A276427, A276428, A276429, A276433, A276434, A277099, A277100, A277101.
%K A277102 nonn
%O A277102 0,4
%A A277102 _Emeric Deutsch_, Oct 10 2016