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%I #33 Oct 10 2017 05:51:29
%S 1,1,2,1,1,2,1,3,3,4,1,2,3,3,4,1,4,5,6,6,7,1,2,5,5,6,6,7,1,5,7,10,10,
%T 11,11,12,1,3,7,8,10,10,11,11,12,1,6,9,14,15,17,17,18,18,19,1,3,9,11,
%U 15,15,17,17,18,18,19,1,7,12,20,22,26,26,28,28,29,29,30
%N Triangle read by rows: T(n,k) = number of palindromic partitions of n in which no part exceeds k, 1 <= k <= n.
%C See A025065 for a definition of palindromic partition.
%H Andrew Howroyd, <a href="/A233322/b233322.txt">Table of n, a(n) for n = 1..1275</a>
%F T(n,k) = Sum_{i=1..k} A233321(n,i).
%F T(n,k) = Sum_{i=0..(k+2*floor(n/2)-n)/2} A026820(floor(n/2)-i, k). - _Andrew Howroyd_, Oct 09 2017
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 1, 2;
%e 1, 3, 3, 4;
%e 1, 2, 3, 3, 4;
%e 1, 4, 5, 6, 6, 7;
%e 1, 2, 5, 5, 6, 6, 7;
%e 1, 5, 7, 10, 10, 11, 11, 12;
%e 1, 3, 7, 8, 10, 10, 11, 11, 12;
%e 1, 6, 9, 14, 15, 17, 17, 18, 18, 19;
%e 1, 3, 9, 11, 15, 15, 17, 17, 18, 18, 19;
%e 1, 7, 12, 20, 22, 26, 26, 28, 28, 29, 29, 30;
%e ...
%t (* run this first: *)
%t Needs["Combinatorica`"];
%t (* run the following in a different cell: *)
%t a233321[n_] := {}; ; Do[Do[a = Partitions[n]; count = 0; Do[If[Max[a[[j]]] == k, x = Permutations[a[[j]]]; Do[If[x[[m]] == Reverse[x[[m]]], count++; Break[]], {m, Length[x]}]], {j, Length[a]}]; AppendTo[a233321[n], count], {k, n}], {n, nmax}]; a233322[n_] := {}; Do[Do[AppendTo[a233322[n], Sum[a233321[n][[j]], {j, k}]], {k, n}], {n,nmax}]; Table[a233322[n], {n, nmax}](* _L. Edson Jeffery_, Oct 09 2017 *)
%o (PARI) \\ here PartitionCount is A026820.
%o PartitionCount(n,maxpartsize)={my(t=0); forpart(p=n, t++, maxpartsize); t}
%o T(n,k)=sum(i=0, (k-n%2)\2, PartitionCount(n\2-i, k));
%o for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ _Andrew Howroyd_, Oct 09 2017
%Y Cf. A025065, A026820; partial sums of row entries of A233321.
%Y Cf. A233323, A233324 (palindromic compositions of n).
%K nonn,tabl
%O 1,3
%A _L. Edson Jeffery_, Dec 10 2013
%E Corrected row 7 as communicated by _Andrew Howroyd_. - _L. Edson Jeffery_, Oct 09 2017
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