A231344 Number of morphisms in full subcategories of Set spanned by {{}, {1}, {1, 2}, ..., {1, 2, ..., n}}.

0, 1, 3, 11, 60, 499, 5705, 82207, 1419768, 28501125, 651233671, 16676686707, 472883844004, 14705395791319, 497538872883741, 18193397941038751, 714950006521386992, 30046260016074301961, 1344648068888240941035, 63841580906470188418243, 3205037832959611351099164

Offset

-1,3

Comments

For any natural number k, consider the set X_k = {1,2,...,k}; in particular X_0 is empty. For any natural number n, let S_n be the full subcategory of Set spanned by the objects X_0, X_1,...,X_n. Then S_n has some number of morphisms, #S_n. When n = -1, we consider S_n to be empty. Our sequence is #S_{-1}, #S_0, #S_1, #S_2,....

Links

Robert G. Wilson v, Table of n, a(n) for n = -1..386

Wikipedia, Category of sets.

Formula

a(n) = Sum_{0 <= i,j <= n} i^j, where 0^0 = 1.

Example

For n=2, we have 0^0+0^1+0^2+1^0+1^1+1^2+2^0+2^1+2^2=11.

Maple

a := n -> add(add(j^k, j = 0..n), k = 0..n):
seq(a(n), n = -1..19);  # Peter Luschny, Dec 24 2025

Mathematica

a[n_] := 1 + Sum[i^j, {j, 0, n}, {i, n}]; a[-1] = 0; Array[a, 19, -1] (* Robert G. Wilson v, Feb 18 2014 *)

Prog

(PARI) a(n)=sum(i=0, n, sum(j=0, n, i^j)) \\ M. F. Hasler, Nov 08 2013

Crossrefs

Cf. A086787.

Sequence in context: A136440 A303871 A381943 * A007146 A076475 A354417

Adjacent sequences: A231341 A231342 A231343 * A231345 A231346 A231347

Keyword

nonn,easy

Author

David Spivak, Nov 07 2013

Extensions

More terms from M. F. Hasler, Nov 08 2013

Status

approved