A319471 -id:A319471 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Sep 19 2018

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

			Triangle T(n,k) begins:
  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,  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, 4, 10, 11, 10,  7, 5, 3, 2, 1, 1;
  0, 0, 5, 11, 15, 13, 11, 7, 5, 3, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A136522 (for n>0), A319468, A261131, A319469, A319470, A319471, A319472, A319473, A319474, A319475.
Row sums give A091580.
T(2n,n) gives A319454.
Cf. A002113, A008284, A072233, A261132.

p:= proc(n) option remember; local i, s; s:= ""||n;
      for i to iquo(length(s), 2) do if
        s[i]<>s[-i] then return false fi od; true
    end:
h:= proc(n) option remember; `if`(n<1, 0,
     `if`(p(n), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..14);

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

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

A341159 Number of partitions of n into 6 distinct nonzero decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 9, 11, 13, 14, 15, 16, 16, 15, 15, 14, 13, 12, 11, 11, 11, 12, 12, 14, 15, 17, 18, 20, 20, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 22, 23, 23, 23, 24, 24, 25, 26, 28, 29, 31, 32, 35, 36, 38, 39, 40, 41, 41, 42, 42, 42, 41, 43, 42, 42, 43, 43, 44, 45, 47

Offset: 21

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 21..20000

Cf. A002113, A136522, A319471, A341155, A341156, A341157, A341158.

Table[Count[IntegerPartitions[n,{6}],?(Length[Union[#]]==Length[#]&&AllTrue[ #,PalindromeQ]&)],{n,21,100}] (* _Harvey P. Dale, Jul 27 2021 *)

A341203 Number of ways to write n as an ordered sum of 6 nonzero decimal palindromes.

Original entry on oeis.org

1, 6, 21, 56, 126, 252, 462, 792, 1287, 1996, 2973, 4272, 5942, 8022, 10536, 13488, 16857, 20592, 24622, 28836, 33108, 37298, 41262, 44862, 47976, 50508, 52398, 53612, 54192, 54186, 53688, 52818, 51708, 50492, 49296, 48228, 47383, 46788, 46497, 46506, 46773, 47268, 47948

Offset: 6

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Cf. A002113, A136522, A319471, A341159, A341184, A341191, A341192, A341193, A341204, A341205, A341206, A341207.

nmax = 48; CoefficientList[Series[Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

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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.

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A341159 Number of partitions of n into 6 distinct nonzero decimal palindromes.

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A341203 Number of ways to write n as an ordered sum of 6 nonzero decimal palindromes.

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