A319453 -id:A319453 - cp's OEIS frontend

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

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, 200, 244, 286, 341, 395, 460, 523, 600, 671, 756, 835, 926, 1008, 1103, 1185, 1280, 1360, 1450, 1524, 1609, 1673, 1748, 1803, 1867, 1910, 1965, 1996, 2039, 2063, 2096

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=9 of A319453.

Cf. A002113.

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, t) option remember; `if`(n=0, 1, `if`(t*i (k-> b(n, h(n), k)-b(n, h(n), k-1))(9):
seq(a(n), n=0..100);

Table[Count[IntegerPartitions[n,{9}],?(AllTrue[#,PalindromeQ]&)],{n,0,60}] (* _Harvey P. Dale, May 24 2025 *)

a(n) = [x^n y^9] 1/Product_{j>=2} (1-y*x^A002113(j)).

A319475 Number of partitions of n into exactly ten nonzero decimal palindromes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 52, 70, 88, 114, 141, 178, 215, 265, 315, 380, 445, 526, 606, 705, 801, 916, 1027, 1159, 1282, 1427, 1561, 1715, 1855, 2015, 2157, 2318, 2458, 2614, 2747, 2897, 3018, 3157, 3266, 3390, 3485

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=10 of A319453.

Cf. A002113.

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, t) option remember; `if`(n=0, 1, `if`(t*i (k-> b(n, h(n), k)-b(n, h(n), k-1))(10):
seq(a(n), n=0..100);

a(n) = [x^n y^10] 1/Product_{j>=2} (1-y*x^A002113(j)).

A319454 Number of partitions of 2n into exactly n nonzero decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 72, 92, 121, 153, 197, 245, 310, 381, 475, 579, 711, 858, 1043, 1248, 1501, 1783, 2126, 2507, 2966, 3476, 4083, 4757, 5551, 6433, 7464, 8606, 9931, 11398, 13089, 14957, 17099, 19461, 22153, 25120, 28483, 32183, 36361

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1500

Cf. A002113, A319453.

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, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> `if`(n=0, 1, b(2*n, h(2*n), n)-b(2*n, h(2*n), n-1)):
seq(a(n), n=0..70);

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

a(n) = A319453(2n,n).

cp's OEIS Frontend

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

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A319475 Number of partitions of n into exactly ten nonzero decimal palindromes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A319454 Number of partitions of 2n into exactly n nonzero decimal palindromes.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Formula