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A340508 Let ped(n) denote the number of partitions of n in which the even parts are distinct (A001935); a(n) = ped(9*n+7).

%I A340508 #30 Jan 29 2021 08:13:12
%S A340508 12,132,876,4416,18624,69060,232044,720648,2097612,5781120,15203904,
%T A340508 38387556,93503052,220586244,505673280,1129518564,2464116480,
%U A340508 5260683840,11010018840,22623235620,45700246668,90863466372,178000194348,343888491684,655760533632,1235186054724
%N A340508 Let ped(n) denote the number of partitions of n in which the even parts are distinct (A001935); a(n) = ped(9*n+7).
%C A340508 These are the coefficients in the left-hand side of a "surprising identity" [Hirschhorn].
%D A340508 M. D. Hirschhorn, The Power of q, Springer, 2017. See (33.1.3) page 303.
%H A340508 Alois P. Heinz, <a href="/A340508/b340508.txt">Table of n, a(n) for n = 0..5000</a>
%F A340508 a(n) = 12 * A226034(n).
%p A340508 with(numtheory):
%p A340508 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A340508      `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)
%p A340508     end:
%p A340508 a:= n-> b(9*n+7):
%p A340508 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 26 2021
%t A340508 b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Sum[
%t A340508    If[Mod[d, 4] == 0, 0, d], {d, Divisors[j]}], {j, 1, n}]/n];
%t A340508 a[n_] := b[9n+7];
%t A340508 a /@ Range[0, 25] (* _Jean-François Alcover_, Jan 29 2021, after _Alois P. Heinz_ *)
%Y A340508 A subsequence of A001935.
%Y A340508 Cf. A226034.
%K A340508 nonn
%O A340508 0,1
%A A340508 _N. J. A. Sloane_, Jan 26 2021