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%I #18 May 17 2023 11:30:06
%S 0,0,0,0,1,1,2,3,5,8,11,16,23,32,45,60,81,109,144,190,247,320,412,529,
%T 675,854,1078,1355,1695,2117,2626,3251,4010,4932,6047,7394,9012,10959,
%U 13290,16083,19407,23379,28090,33689,40317,48158,57406,68324,81155,96248
%N Number of partitions p of n such that 4*min(p) is a part of p.
%H Alois P. Heinz, <a href="/A238591/b238591.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: Sum_{k>=1} x^(5*k)/Product_{j>=k} (1-x^j). - _Seiichi Manyama_, May 17 2023
%e a(9) = 5 counts these partitions: 441, 4311, 4221, 42111, 411111.
%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-5*i, i), i=1..n/5):
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Mar 03 2014
%t Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*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 - 5*i, i], {i, 1, n/5}];
%t Array[a, 60] (* _Jean-François Alcover_, Jun 09 2021, after _Alois P. Heinz_ *)
%o (PARI) my(N=50, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, N, 1-x^j)))) \\ _Seiichi Manyama_, May 17 2023
%Y Cf. A117989, A238589, A238590.
%Y Cf. A237826, A363067.
%K nonn,easy
%O 1,7
%A _Clark Kimberling_, Mar 01 2014
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