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%I #51 Jul 10 2023 20:16:50
%S 1,1,2,4,7,12,22,36,60,97,162,254,406,628,974,1514,2305,3492,5254,
%T 7842,11598,17292,25294,37090,53866,78113,112224,161092,230788,328352,
%U 466040,658846,928132,1302290,1821770,2537156,3536445,4897310,6777806,9341456,12858960,17625970,24133832,32910898,44813228,60922160,82569722
%N Number of compositions of n if all summand runs are kept together.
%C a(n^2) is odd. - _Gregory L. Simay_, Jun 23 2019
%C Also the number of compositions of n avoiding the patterns (1,2,1) and (2,1,2). - _Gus Wiseman_, Jul 07 2020
%H Alois P. Heinz, <a href="/A274174/b274174.txt">Table of n, a(n) for n = 0..5000</a>
%H Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%F a(n) = Sum_{k>=0} k! * A116608(n,k). - _Joerg Arndt_, Jun 12 2016
%e If the summand runs are blocked together, there are 22 compositions of a(6): 6; 5+1, 1+5, 4+2, 2+4, (3+3), 4+(1+1), (1+1)+4, 1+2+3, 1+3+2, 2+1+3, 2+3+1, 3+1+2, 3+2+1, (2+2+2), 3+(1+1+1), (1+1+1)+3, (2+2)+(1+1), (1+1)+(2+2), 2+(1+1+1+1), (1+1+1+1)+2, (1+1+1+1+1+1).
%e a(0)=1; a(1)= 1; a(4) = 7; a(9) = 97; a(16) = 2305; a(25) = 78113 and a(36) = 3536445. - _Gregory L. Simay_, Jun 23 19
%p b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p add(b(n-i*j, i-1, p+`if`(j=0, 0, 1)), j=0..n/i)))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jun 12 2016
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==Length[Union[#]]&]],{n,0,10}] (* _Gus Wiseman_, Jul 07 2020 *)
%t b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
%t Sum[b[n - i*j, i - 1, p + If[j == 0, 0, 1]], {j, 0, n/i}]]];
%t a[n_] := b[n, n, 0];
%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 11 2021, after _Alois P. Heinz_ *)
%Y Cf. A000070, A101880, A116608.
%Y The version for patterns is A001339.
%Y The version for prime indices is A333175.
%Y The complement (i.e., the matching version) is A335548.
%Y Anti-run compositions are A003242.
%Y (1,2,1)- and (2,1,2)-matching permutations of prime indices are A335462.
%Y (1,2,1)-matching compositions are A335470.
%Y (1,2,1)-avoiding compositions are A335471.
%Y (2,1,2)-matching compositions are A335472.
%Y (2,1,2)-avoiding compositions are A335473.
%Y Cf. A000670, A056986, A181796, A335451, A335452, A335460, A335463.
%K nonn
%O 0,3
%A _Gregory L. Simay_, Jun 12 2016
%E Terms a(9) and beyond from _Joerg Arndt_, Jun 12 2016
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