A071268 Sum of all digit permutations of the concatenation of first n numbers.

1, 33, 1332, 66660, 3999960, 279999720, 22399997760, 2015999979840, 201599999798400, 927359999990726400, 1064447999999893555200, 2058376319999997941623680, 4439635199999999955603648000, 10585935359999999998941406464000, 27655756127999999999972344243872000

Offset: 1

Author

Amarnath Murthy, Jun 01 2002

Keywords

• base
• nonn

Comments

The permutations yield n! different numbers and if they are stacked vertically then the sum of each column is (n-1)! times the n-th triangular number = (n-1)!*n(n+1)/2. a(n) = [(n+1)!/2]*[{10^n -1}/9]. Note that this is only valid for 1 <= n <= 9.

The first person who studied the sum of different permutations of digits of a given number seems to be the French scientist Eugène Aristide Marre (1823-1918). See links. - Bernard Schott, Dec 07 2012

Examples

			For n=3, a(3) = 123 + 132 + 213 + 231 + 312 + 321 = 1332. - _Michael B. Porter_, Aug 28 2016
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..208, Jan 04 2019
• A. Marre, Trouver la somme de toutes les permutations différentes d'un nombre donné., Nouvelles Annales de Mathématiques, 1ère série, tome 5 (1846), p. 57-60.
• Norbert Verdier and Raymond Cordier, QDV4 : Marre, Marre et Marre, page=1 (French mathematical forum les-mathematiques.net)

Crossrefs

Cf. A045876, A047726, A007908.

Programs

Maple

a:= proc(n) local s, t, l;
      s:= cat("", seq(i, i=1..n)); t:= length(s);
      l:= (p-> seq(coeff(p, x, i), i=0..9))(add(x^parse(s[i]), i=1..t));
      (10^t-1)/9*combinat[multinomial](t, l)*add(i*l[i+1], i=1..9)/t
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 04 2019

Mathematica

Table[Total@ Map[FromDigits, Permutations@ Flatten@ Map[IntegerDigits, Range@ n]], {n, 10}] (* or *)
Table[Function[d, (((10^Length@ d - 1)/9)* Length@ Union@ Map[FromDigits, Permutations@ d] Total[d])/Length@ d]@ Flatten@ Map[IntegerDigits, Range@ n], {n, 11}] (* Michael De Vlieger, Aug 30 2016, latter after Harvey P. Dale at A047726 *)

PARI

A007908(n) = my(s=""); for(k=1, n, s=Str(s, k)); eval(s);
A047726(n) = n=eval(Vec(Str(n))); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
a(n) = ((10^A055642(A007908(n))-1)/9)*(A047726(A007908(n))*A007953(A007908(n))/(A055642(A007908(n)))); \\ Altug Alkan, Aug 28 2016

Python

from math import factorial
from operator import mul
from functools import reduce
def A071268(n):
    s = ''.join(str(i) for i in range(1,n+1))
    return sum(int(d) for d in s)*factorial(len(s)-1)*(10**len(s)-1)//(9*reduce(mul,(factorial(d) for d in (s.count(w) for w in set(s))))) # Chai Wah Wu, Jan 04 2019

Formula

a(n) = (n + 1)!*(10^n - 1)/18 for 1 <= n <= 9.

a(n) = ((10^A055642(A007908(n))-1)/9)*(A047726(A007908(n))*A007953(A007908(n))/(A055642(A007908(n)))). - Altug Alkan, Aug 28 2016

Extensions

Edited by Robert G. Wilson v, Jun 03 2002

Corrected by Altug Alkan, Aug 28 2016