A319453 - OEIS

%I #23 Aug 19 2021 18:24:34

%S 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 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 4,10,11,10,7,5,3,2,1,1,0,0,5,11,15,13,11,7,5,3,2,1,1

%N 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 Differs from A008284 and from A072233 first at T(10,1) = 0.

%H Alois P. Heinz, <a href="/A319453/b319453.txt">Rows n = 0..200, flattened</a>

%F T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A002113(j)).

%F Sum_{k=0..3} T(n,k) = A261132(n).

%e Triangle T(n,k) begins:

%e   1;

%e   0, 1;

%e   0, 1, 1;

%e   0, 1, 1,  1;

%e   0, 1, 2,  1,  1;

%e   0, 1, 2,  2,  1,  1;

%e   0, 1, 3,  3,  2,  1,  1;

%e   0, 1, 3,  4,  3,  2,  1, 1;

%e   0, 1, 4,  5,  5,  3,  2, 1, 1;

%e   0, 1, 4,  7,  6,  5,  3, 2, 1, 1;

%e   0, 0, 5,  8,  9,  7,  5, 3, 2, 1, 1;

%e   0, 1, 4, 10, 11, 10,  7, 5, 3, 2, 1, 1;

%e   0, 0, 5, 11, 15, 13, 11, 7, 5, 3, 2, 1, 1;

%e   ...

%p p:= proc(n) option remember; local i, s; s:= ""||n;

%p       for i to iquo(length(s), 2) do if

%p         s[i]<>s[-i] then return false fi od; true

%p     end:

%p h:= proc(n) option remember; `if`(n<1, 0,

%p      `if`(p(n), n, h(n-1)))

%p     end:

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,

%p       b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):

%p seq(T(n), n=0..14);

%Y Columns k=0-10 give: A000007, A136522 (for n>0), A319468, A261131, A319469, A319470, A319471, A319472, A319473, A319474, A319475.

%Y Row sums give A091580.

%Y T(2n,n) gives A319454.

%Y Cf. A002113, A008284, A072233, A261132.

%K nonn,tabl,base

%O 0,13

%A _Alois P. Heinz_, Sep 19 2018