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A319453 Number T(n,k) of partitions of n into exactly k nonzero decimal palindromes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A319453 #23 Aug 19 2021 18:24:34
%S A319453 1,0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,2,2,1,1,0,1,3,3,2,1,1,0,1,3,4,3,2,
%T A319453 1,1,0,1,4,5,5,3,2,1,1,0,1,4,7,6,5,3,2,1,1,0,0,5,8,9,7,5,3,2,1,1,0,1,
%U A319453 4,10,11,10,7,5,3,2,1,1,0,0,5,11,15,13,11,7,5,3,2,1,1
%N A319453 Number T(n,k) of partitions of n into exactly k nonzero decimal palindromes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A319453 Differs from A008284 and from A072233 first at T(10,1) = 0.
%H A319453 Alois P. Heinz, <a href="/A319453/b319453.txt">Rows n = 0..200, flattened</a>
%F A319453 T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A002113(j)).
%F A319453 Sum_{k=0..3} T(n,k) = A261132(n).
%e A319453 Triangle T(n,k) begins:
%e A319453   1;
%e A319453   0, 1;
%e A319453   0, 1, 1;
%e A319453   0, 1, 1,  1;
%e A319453   0, 1, 2,  1,  1;
%e A319453   0, 1, 2,  2,  1,  1;
%e A319453   0, 1, 3,  3,  2,  1,  1;
%e A319453   0, 1, 3,  4,  3,  2,  1, 1;
%e A319453   0, 1, 4,  5,  5,  3,  2, 1, 1;
%e A319453   0, 1, 4,  7,  6,  5,  3, 2, 1, 1;
%e A319453   0, 0, 5,  8,  9,  7,  5, 3, 2, 1, 1;
%e A319453   0, 1, 4, 10, 11, 10,  7, 5, 3, 2, 1, 1;
%e A319453   0, 0, 5, 11, 15, 13, 11, 7, 5, 3, 2, 1, 1;
%e A319453   ...
%p A319453 p:= proc(n) option remember; local i, s; s:= ""||n;
%p A319453       for i to iquo(length(s), 2) do if
%p A319453         s[i]<>s[-i] then return false fi od; true
%p A319453     end:
%p A319453 h:= proc(n) option remember; `if`(n<1, 0,
%p A319453      `if`(p(n), n, h(n-1)))
%p A319453     end:
%p A319453 b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
%p A319453       b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
%p A319453     end:
%p A319453 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
%p A319453 seq(T(n), n=0..14);
%Y A319453 Columns k=0-10 give: A000007, A136522 (for n>0), A319468, A261131, A319469, A319470, A319471, A319472, A319473, A319474, A319475.
%Y A319453 Row sums give A091580.
%Y A319453 T(2n,n) gives A319454.
%Y A319453 Cf. A002113, A008284, A072233, A261132.
%K A319453 nonn,tabl,base
%O A319453 0,13
%A A319453 _Alois P. Heinz_, Sep 19 2018