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A126285 Number of partial mappings (or mapping patterns) from n points to themselves; number of partial endofunctions. 5
1, 2, 6, 16, 45, 121, 338, 929, 2598, 7261, 20453, 57738, 163799, 465778, 1328697, 3798473, 10883314, 31237935, 89812975, 258595806, 745563123, 2152093734, 6218854285, 17988163439, 52078267380, 150899028305, 437571778542, 1269754686051, 3687025215421 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

If an endofunction is partial, then some points may be unmapped (or mapped to "undefined").

The labeled version is left-shifted A000169. - Franklin T. Adams-Watters, Jan 16 2007

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..750

RI McLachlan, K Modin, H Munthe-Kaas, O Verdier, What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations, arXiv preprint arXiv:1512.00906, 2015

N. J. A. Sloane, Transforms

FORMULA

Euler transform of A002861 + A000081 = [1, 2, 4, 9, 20, 51, 125, 329, 862, 2311, ... ] + [ 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, ...] = A124682.

Convolution of left-shifted A000081 with A001372. - Franklin T. Adams-Watters, Jan 16 2007

a(n) ~ c * d^n / sqrt(n), where d = 2.95576528565... is the Otter's rooted tree constant (see A051491) and c = 1.309039781943936352117502717... - Vaclav Kotesovec, Mar 29 2014

MATHEMATICA

nmax = 28;

a81[n_] := a81[n] = If[n<2, n, Sum[Sum[d*a81[d], {d, Divisors[j]}]*a81[n-j ], {j, 1, n-1}]/(n-1)];

A[n_] := A[n] = If[n<2, n, Sum[DivisorSum[j, #*A[#]&]*A[n-j], {j, 1, n-1} ]/(n-1)];

H[t_] := Sum[A[n]*t^n, {n, 0, nmax+2}];

F = 1/Product[1 - H[x^n], {n, 1, nmax+2}] + O[x]^(nmax+2);

A1372 = CoefficientList[F, x];

a[n_] := Sum[a81[k] * A1372[[n-k+2]], {k, 0, n+1}];

Table[a[n], {n, 0, nmax}] (* Jean-Fran├žois Alcover, Aug 18 2018, after Franklin T. Adams-Watters *)

CROSSREFS

Cf. A001372.

Cf. A000169, A000081, A002861.

Sequence in context: A263897 A209629 A055544 * A026163 A005717 A025266

Adjacent sequences:  A126282 A126283 A126284 * A126286 A126287 A126288

KEYWORD

nonn

AUTHOR

Christian G. Bower, Dec 25 2006 based on a suggestion from Jonathan Vos Post

STATUS

approved

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Last modified October 21 06:25 EDT 2018. Contains 316405 sequences. (Running on oeis4.)