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A319474 Number of partitions of n into exactly nine nonzero decimal palindromes.

%I A319474 #8 May 24 2025 13:48:48
%S A319474 0,0,0,0,0,0,0,0,0,1,1,2,3,5,7,11,15,22,29,40,51,68,85,109,134,167,
%T A319474 200,244,286,341,395,460,523,600,671,756,835,926,1008,1103,1185,1280,
%U A319474 1360,1450,1524,1609,1673,1748,1803,1867,1910,1965,1996,2039,2063,2096
%N A319474 Number of partitions of n into exactly nine nonzero decimal palindromes.
%H A319474 Alois P. Heinz, <a href="/A319474/b319474.txt">Table of n, a(n) for n = 0..10000</a>
%F A319474 a(n) = [x^n y^9] 1/Product_{j>=2} (1-y*x^A002113(j)).
%p A319474 p:= proc(n) option remember; local i, s; s:= ""||n;
%p A319474       for i to iquo(length(s), 2) do if
%p A319474         s[i]<>s[-i] then return false fi od; true
%p A319474     end:
%p A319474 h:= proc(n) option remember; `if`(n<1, 0,
%p A319474      `if`(p(n), n, h(n-1)))
%p A319474     end:
%p A319474 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i<n,
%p A319474       0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
%p A319474     end:
%p A319474 a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(9):
%p A319474 seq(a(n), n=0..100);
%t A319474 Table[Count[IntegerPartitions[n,{9}],_?(AllTrue[#,PalindromeQ]&)],{n,0,60}] (* _Harvey P. Dale_, May 24 2025 *)
%Y A319474 Column k=9 of A319453.
%Y A319474 Cf. A002113.
%K A319474 nonn,base
%O A319474 0,12
%A A319474 _Alois P. Heinz_, Sep 19 2018