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A373950 Number of integer compositions of n containing two adjacent ones and no other runs.

%I A373950 #14 Jul 02 2024 02:12:17
%S A373950 0,0,1,0,2,4,5,14,26,46,92,176,323,610,1145,2108,3912,7240,13289,
%T A373950 24418,44778,81814,149356,272222,495144,899554,1632176,2957332,
%U A373950 5352495,9677266,17477761,31536288,56852495,102403134,184302331,331452440,595659234,1069742760
%N A373950 Number of integer compositions of n containing two adjacent ones and no other runs.
%C A373950 Also the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) results in a composition of n-1.
%H A373950 John Tyler Rascoe, <a href="/A373950/b373950.txt">Table of n, a(n) for n = 0..1000</a>
%F A373950 a(n>0) = A373949(n,n-1).
%F A373950 G.f.: x/((1-x)^2 * (1 - Sum_{i>0} (x^i/(1+x^i)))^2). - _John Tyler Rascoe_, Jul 02 2024
%e A373950 The a(0) = 0 through a(7) = 14 compositions:
%e A373950   .  .  (11)  .  (112)  (113)   (114)   (115)
%e A373950                  (211)  (311)   (411)   (511)
%e A373950                         (1121)  (1131)  (1123)
%e A373950                         (1211)  (1311)  (1132)
%e A373950                                 (2112)  (1141)
%e A373950                                         (1411)
%e A373950                                         (2113)
%e A373950                                         (2311)
%e A373950                                         (3112)
%e A373950                                         (3211)
%e A373950                                         (11212)
%e A373950                                         (12112)
%e A373950                                         (21121)
%e A373950                                         (21211)
%t A373950 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==n-1&]],{n,0,10}]
%o A373950 (PARI)
%o A373950 A_x(N)={my(x='x+O('x^N), h=x/((1+x)^2*(1-sum(i=1,N, (x^i /(1+x^i))))^2)); concat([0, 0], Vec(h))}
%o A373950 A_x(40) \\ _John Tyler Rascoe_, Jul 02 2024
%Y A373950 For any run (not just of ones) we have A003242.
%Y A373950 Subdiagonal of A373949.
%Y A373950 These compositions are ranked by A373956.
%Y A373950 A003242 counts compressed compositions.
%Y A373950 A114901 counts compositions with no isolated parts.
%Y A373950 A116861 counts partitions by compressed sum, by compressed length A116608.
%Y A373950 A333755 counts compositions by compressed length (number of runs).
%Y A373950 A373948 represents the run-compression transformation.
%Y A373950 Cf. A106356, A238130, A238279, A238343, A240085, A285981, A333213, A333489, A373951, A373952.
%K A373950 nonn
%O A373950 0,5
%A A373950 _Gus Wiseman_, Jun 28 2024
%E A373950 a(26) onwards from _John Tyler Rascoe_, Jul 02 2024