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Number of partitions of n into exactly five nonzero decimal palindromes.
4

%I #11 Oct 20 2024 10:03:28

%S 0,0,0,0,0,1,1,2,3,5,7,10,13,18,22,29,34,42,48,57,63,72,77,85,88,95,

%T 96,100,99,101,98,98,93,92,87,85,80,79,74,73,70,69,67,67,65,66,65,66,

%U 66,67,68,69,70,72,73,75,76,78,79,81,81,83,83,84,84,85,84

%N Number of partitions of n into exactly five nonzero decimal palindromes.

%H Alois P. Heinz, <a href="/A319470/b319470.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = [x^n y^5] 1/Product_{j>=2} (1-y*x^A002113(j)).

%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, t) option remember; `if`(n=0, 1, `if`(t*i<n,

%p 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))

%p end:

%p a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(5):

%p seq(a(n), n=0..100);

%t Table[Count[IntegerPartitions[n,{5}],_?(AllTrue[#,PalindromeQ]&)],{n,0,70}] (* _Harvey P. Dale_, Oct 20 2024 *)

%Y Column k=5 of A319453.

%Y Cf. A002113.

%K nonn,base

%O 0,8

%A _Alois P. Heinz_, Sep 19 2018