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A327710 Number of compositions of n into distinct parts such that the difference between any two parts is at least two.

%I A327710 #38 Jul 12 2021 13:48:23
%S A327710 1,1,1,1,3,3,5,5,7,13,15,21,29,35,43,55,87,99,137,173,235,277,363,429,
%T A327710 545,755,895,1135,1443,1827,2285,2837,3463,4285,5199,6309,8237,9755,
%U A327710 12091,14743,18351,22251,27833,33125,40819,49045,59691,70869,86033,106163
%N A327710 Number of compositions of n into distinct parts such that the difference between any two parts is at least two.
%C A327710 All terms are odd.
%H A327710 Alois P. Heinz, <a href="/A327710/b327710.txt">Table of n, a(n) for n = 0..10000</a>
%F A327710 a(n) = Sum_{k>=0} k! * A268187(n,k).
%F A327710 G.f.: Sum_{k>=0} k! * x^(k^2) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Dec 04 2020
%e A327710 a(9) = 13: 135, 153, 315, 351, 513, 531, 36, 63, 27, 72, 18, 81, 9.
%p A327710 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A327710      `if`(n=0, p!, b(n, i-1, p)+b(n-i, min(n-i, i-2), p+1)))
%p A327710     end:
%p A327710 a:= n-> b(n$2, 0):
%p A327710 seq(a(n), n=0..50);
%p A327710 # second Maple program:
%p A327710 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A327710      `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
%p A327710     end:
%p A327710 a:= n-> add(k!*b(n-k^2, k), k=0..floor(sqrt(n))):
%p A327710 seq(a(n), n=0..50);
%t A327710 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A327710      b[n, i - 1] + b[n - i, Min[n - i, i]]]];
%t A327710 a[n_] := Sum[k!*b[n - k^2, k], {k, 0, Floor[Sqrt[n]]}];
%t A327710 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 12 2021, after _Alois P. Heinz_ *)
%Y A327710 Cf. A003114, A032020, A268187, A268188, A328222.
%K A327710 nonn
%O A327710 0,5
%A A327710 _Alois P. Heinz_, Feb 24 2020