cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A347420 Number of partitions of [n] where the first k elements are marked (0 <= k <= n) and at least k blocks contain their own index.

Original entry on oeis.org

1, 2, 5, 14, 45, 164, 667, 2986, 14551, 76498, 430747, 2582448, 16403029, 109918746, 774289169, 5715471606, 44087879137, 354521950932, 2965359744447, 25749723493074, 231719153184019, 2157494726318234, 20753996174222511, 205985762120971168, 2106795754056142537
Offset: 0

Views

Author

Alois P. Heinz, Jan 05 2022

Keywords

Examples

			a(3) = 14 = 5 + 5 + 3 + 1: 123, 12|3, 13|2, 1|23, 1|2|3, 1'23, 1'2|3, 1'3|2, 1'|23, 1'|2|3, 1'3|2', 1'|2'3, 1'|2'|3, 1'|2'|3'.
		

Crossrefs

Antidiagonal sums of A108087.

Programs

  • Maple
    b:= proc(n, m) option remember;
         `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
        end:
    a:= n-> add(b(i, n-i), i=0..n):
    seq(a(n), n=0..25);
  • Mathematica
    b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m + 1] + m*b[n - 1, m]];
    a[n_] := Sum[b[i, n - i], {i, 0, n}];
    Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} A108087(n-k,k).
a(n) = 1 + A005490(n).
a(n) = A000110(n) + Sum_{k=1..n} k * A259691(n-1,k).
a(n) = Sum_{k=1..n} (k+1) * A259691(n-1,k).
a(n) = A000110(n) + A350589(n).
a(n) mod 2 = A059841(n).