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A369584 a(n) = 2^(n - 3) - A368279(n) for n >= 5, otherwise 0. Number of compositions of n whose first and last part is not equal to 1 and whose first part is not the largest part.

%I A369584 #21 Apr 13 2025 07:11:18
%S A369584 0,0,0,0,0,1,2,6,13,30,65,140,296,622,1294,2679,5518,11323,23160,
%T A369584 47250,96184,195438,396490,803292,1625591,3286340,6637913,13397224,
%U A369584 27020974,54465702,109725932,220944768,444700208,894701728,1799419458,3617792587,7271505058
%N A369584 a(n) = 2^(n - 3) - A368279(n) for n >= 5, otherwise 0. Number of compositions of n whose first and last part is not equal to 1 and whose first part is not the largest part.
%C A369584 We consider a tripartition of the compositions of an integer n >= 1, which we sort in lexicographic order. For this purpose, we define three predicates for a composition C.
%C A369584   (1) The first part of C is the largest part of C.
%C A369584   (2) The first part of C is 1.
%C A369584   (3) The last part of C is 1.
%C A369584 We thus define three classes of compositions:
%C A369584   X(n): compositions for which (1) and not (3) applies;
%C A369584   Y(n): compositions for which (2) or (3) applies;
%C A369584   Z(n): compositions for which not (1) and not (2) and not (3) applies.
%C A369584 X(n), Y(n), and Z(n) are disjoint classes whose union comprises all compositions of n; they thus form a disjoint tripartition of the compositions of n. The number of these compositions are given by:
%C A369584   card(Z(n)) = a(n); card(X(n)) = A368279(n); card(Y(n)) = A122391(n).
%F A369584 a(n) = [x^n] (1 + (x - 1)*(1 + x^2/(2*x - 1) + Sum_{k>=1} x^k/(1 - x*(1 - x^k)/(1 - x)))).
%F A369584 card(X(n)) + card(Y(n)) + card(Z(n)) = A011782(n) = 2^(n - 1) for n > 0.
%e A369584 The compositions of class Z(n) for n = 5..7 are:
%e A369584   5: [2, 3];
%e A369584   6: [2, 1, 3], [2, 4];
%e A369584   7: [2, 1, 1, 3], [2, 1, 4], [2, 2, 3], [2, 3, 2], [2, 5], [3, 4].
%e A369584 The cardinalities of some tripartitions, |X(n)| + |Y(n)| + |Z(n)| = 2^(n - 1):
%e A369584    5:  12 +  3 +  1 =  16;
%e A369584    6:  24 +  6 +  2 =  32;
%e A369584    7:  48 + 10 +  6 =  64;
%e A369584    8:  96 + 19 + 13 = 128;
%e A369584    9: 192 + 34 + 30 = 256;
%e A369584   10: 384 + 63 + 65 = 512.
%p A369584 gf := 1 + (x - 1)*(1 + x^2 / (2*x - 1) + sum(x^n / (1 - x*(1 - x^n)/(1 - x)),
%p A369584 n = 1..42)): ser := series(gf, x, 40): seq(coeff(ser, x, n), n = 0..36);
%o A369584 (SageMath)
%o A369584 def A369584(n):
%o A369584     if n < 5: return 0
%o A369584     return 2^(n - 3) - A368279(n)
%o A369584 print([A369584(n) for n in range(37)])
%Y A369584 Cf. A011782, A122391, A368279.
%K A369584 nonn
%O A369584 0,7
%A A369584 _Peter Luschny_, Jan 29 2024