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%I #16 May 17 2023 11:30:10
%S 0,0,0,1,1,2,3,6,7,12,16,25,32,46,61,86,110,149,192,257,326,425,538,
%T 694,871,1107,1381,1740,2154,2689,3313,4103,5024,6176,7529,9201,11157,
%U 13554,16365,19784,23782,28610,34260,41039,48958,58405,69431,82525,97775
%N Number of partitions p of n such that 3*min(p) is a part of p.
%H Alois P. Heinz, <a href="/A238590/b238590.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: Sum_{k>=1} x^(4*k)/Product_{j>=k} (1-x^j). - _Seiichi Manyama_, May 17 2023
%e a(7) = 3 counts these partitions: 331, 3211, 31111.
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
%p end:
%p a:= n-> add(b(n-4*i, i), i=1..n/4):
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Mar 03 2014
%t Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*Min[p]]], {n, 50}]
%t (* Second program: *)
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
%t a[n_] := Sum[b[n-4i, i], {i, 1, n/4}];
%t Array[a, 60] (* _Jean-François Alcover_, Jun 04 2021, after _Alois P. Heinz_ *)
%o (PARI) my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, N, 1-x^j)))) \\ _Seiichi Manyama_, May 17 2023
%Y Cf. A117989, A238589, A238591.
%Y Cf. A237825, A363066.
%K nonn,easy
%O 1,6
%A _Clark Kimberling_, Mar 01 2014
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