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A383895 Number of spiny partitions with exactly n parts.

%I A383895 #20 Jun 30 2025 16:00:08
%S A383895 1,1,2,4,9,20,47,111,267,646,1582,3892,9636,23961,59871,150128,377738,
%T A383895 953029,2410626,6111055,15524013,39508683,100719223,257150952,
%U A383895 657454544,1683042629,4313582090,11067748352,28426813910,73082880708,188059428289,484330230117,1248338233493
%N A383895 Number of spiny partitions with exactly n parts.
%C A383895 An integer partition is said to be spiny if for all parts k having multiplicity m the number of parts <= k is >= m*k.
%C A383895 Arborescent partitions (cf. A383894) are spiny partitions.
%H A383895 Alois P. Heinz, <a href="/A383895/b383895.txt">Table of n, a(n) for n = 0..200</a>
%e A383895 The 20 spiny partitions corresponding to a(5) = 20 are:
%e A383895   (11111),  (21111),  (22111),  (31111),  (32111),
%e A383895   (32211),  (41111),  (42111),  (42211),  (43111),
%e A383895   (43211),  (51111),  (52111),  (52211),  (53111),
%e A383895   (53211),  (54111),  (54211),  (54311),  (54321).
%e A383895 The partition (42221) is not spiny because the part 2 has multiplicity 3 but the number of parts <=2 is 4 < 3*2.
%e A383895 The only spiny partition of length 5 which does not correspond to an arborescent partition is (42211), i.e. there is no tree whose multiset of subtree sizes is {6, 4, 2, 2, 1, 1} (cf. A383894).
%o A383895 (Python)
%o A383895 def A383895(n): #generator of terms a(0) to a(n)
%o A383895     L = [[1]]
%o A383895     for k in range(1,n+2):
%o A383895         l = [0]
%o A383895         for i in range(1,k+1):
%o A383895             l.append(sum(L[a][b] for a in range(k-(k//i),k) for b in range(i)))
%o A383895         L.append(l)
%o A383895         yield l[-1]
%Y A383895 Cf. A382440, A383894.
%K A383895 nonn
%O A383895 0,3
%A A383895 _Ludovic Schwob_, May 14 2025