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A368279 a(n) is the number of compositions of n where the first part is the largest part and the last part is not 1. Row sums of A368579.

%I A368279 #39 Feb 27 2024 07:17:11
%S A368279 1,0,1,1,2,3,6,10,19,34,63,116,216,402,754,1417,2674,5061,9608,18286,
%T A368279 34888,66706,127798,245284,471561,907964,1750695,3379992,6533458,
%U A368279 12643162,24491796,47490688,92170704,179040096,348064190,677174709,1318429534,2568691317
%N A368279 a(n) is the number of compositions of n where the first part is the largest part and the last part is not 1. Row sums of A368579.
%C A368279 Considering more generally the family of generating functions (1 - x)^n * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)) one finds several sequences related to compositions as indicated in the cross-references.
%C A368279 The compositions considered here can also be understood as perfectly balanced, ordered trees. See the linked illustrations. - _Peter Luschny_, Feb 26 2024
%H A368279 Peter Luschny, <a href="/A368279/a368279.txt">A generator for the A368279 compositions</a>.
%H A368279 Peter Luschny, <a href="/A368279/a368279.pdf">Some perfectly balanced, ordered trees illustrating A368279</a>.
%F A368279 a(n) = Sum_{k=0..n} (F(k+1, n+1-k) - F(k+1, n-k)) where F(k, n) = Sum_{j=1..min(k, n)} F(k, n-j) if n > 1 and otherwise n. F(k, n) refers to the generalized Fibonacci number A092921.
%F A368279 a(n) = A007059(n+1) - A007059(n).
%F A368279 G.f.: (1 - x)*(Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k ))) = (1 - x) * GfA079500. - _Peter Luschny_, Jan 20 2024
%e A368279 a(0) = card({[0]}) = 1.
%e A368279 a(1) = card({}) = 0.
%e A368279 a(2) = card({[2]}) = 1.
%e A368279 a(3) = card({[3]}) = 1.
%e A368279 a(4) = card({[2, 2], [4]}) = 2.
%e A368279 a(5) = card({[2, 1, 2], [3, 2], [5]}) = 3.
%e A368279 a(6) = card({[2, 2, 2], [2, 1, 1, 2], [3, 3], [3, 1, 2], [4, 2], [6]}) = 6.
%e A368279 a(7) = card({[2, 2, 1, 2], [2, 1, 2, 2], [2, 1, 1, 1, 2], [3, 2, 2], [3, 1, 3], [3, 1, 1, 2], [4, 3], [4, 1, 2], [5, 2], [7]}) = 10.
%e A368279 a(8) = card({[2, 2, 2, 2],  [2, 2, 1, 1, 2], [2, 1, 2, 1, 2], [2, 1, 1, 2, 2], [2, 1, 1, 1, 1, 2], [3, 3, 2], [3, 2, 3], [3, 2, 1, 2], [3, 1, 2, 2], [3, 1, 1, 3], [3, 1, 1, 1, 2], [4, 4], [4, 2, 2], [4, 1, 3], [4, 1, 1, 2], [5, 3], [5, 1, 2], [6, 2], [8]}) = 19.
%p A368279 gf := (1 - x)*sum(x^j / (1 - sum(x^k, k = 1..j)), j = 0..42):
%p A368279 ser := series(gf, x, 40): seq(coeff(ser, x, n), n = 0..37);
%p A368279 # _Peter Luschny_, Jan 19 2024
%o A368279 (Python)
%o A368279 from functools import cache
%o A368279 @cache
%o A368279 def F(k, n):
%o A368279     return sum(F(k,n-j) for j in range(1,min(k,n))) if n>1 else n
%o A368279 def a(n): return sum(F(k+1, n+1-k) - F(k+1, n-k) for k in range(n+1))
%o A368279 print([a(n) for n in range(38)])
%o A368279 (SageMath)
%o A368279 def C(n): return sum(Compositions(n, max_part=k, inner=[k]).cardinality()
%o A368279                  for k in (0..n))
%o A368279 def a(n): return C(n) - C(n-1) if n > 1 else 1 - n
%o A368279 print([a(n) for n in (0..28)])
%Y A368279 Cf. A368579, A369492, A007059, A092921, A188541.
%Y A368279 Cf. A369115 (n=-2), A186537 left shifted (n=-1), A079500 (n=0), this sequence (n=1), A369116 (n=2).
%K A368279 nonn
%O A368279 0,5
%A A368279 _Peter Luschny_, Jan 04 2024