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A319470 Number of partitions of n into exactly five nonzero decimal palindromes.

%I A319470 #11 Oct 20 2024 10:03:28
%S A319470 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 A319470 96,100,99,101,98,98,93,92,87,85,80,79,74,73,70,69,67,67,65,66,65,66,
%U A319470 66,67,68,69,70,72,73,75,76,78,79,81,81,83,83,84,84,85,84
%N A319470 Number of partitions of n into exactly five nonzero decimal palindromes.
%H A319470 Alois P. Heinz, <a href="/A319470/b319470.txt">Table of n, a(n) for n = 0..10000</a>
%F A319470 a(n) = [x^n y^5] 1/Product_{j>=2} (1-y*x^A002113(j)).
%p A319470 p:= proc(n) option remember; local i, s; s:= ""||n;
%p A319470       for i to iquo(length(s), 2) do if
%p A319470         s[i]<>s[-i] then return false fi od; true
%p A319470     end:
%p A319470 h:= proc(n) option remember; `if`(n<1, 0,
%p A319470      `if`(p(n), n, h(n-1)))
%p A319470     end:
%p A319470 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i<n,
%p A319470       0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
%p A319470     end:
%p A319470 a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(5):
%p A319470 seq(a(n), n=0..100);
%t A319470 Table[Count[IntegerPartitions[n,{5}],_?(AllTrue[#,PalindromeQ]&)],{n,0,70}] (* _Harvey P. Dale_, Oct 20 2024 *)
%Y A319470 Column k=5 of A319453.
%Y A319470 Cf. A002113.
%K A319470 nonn,base
%O A319470 0,8
%A A319470 _Alois P. Heinz_, Sep 19 2018