A145126 a(n) = 1 + (6 + (11 + (6 + n)*n)*n)*n/24.

1, 2, 6, 16, 36, 71, 127, 211, 331, 496, 716, 1002, 1366, 1821, 2381, 3061, 3877, 4846, 5986, 7316, 8856, 10627, 12651, 14951, 17551, 20476, 23752, 27406, 31466, 35961, 40921, 46377, 52361, 58906, 66046, 73816, 82252, 91391, 101271, 111931, 123411, 135752

Offset: 0

Author

Alois P. Heinz, Oct 03 2008

Keywords

• nonn
• easy

Comments

From Gary W. Adamson, Jul 31 2010: (Start)

Equals (1, 2, 3, 4, 5, ...) convolved with (1, 0, 3, 6, 10, 15, ...).

Example: a(4) = 36 = (5, 4, 3, 2, 1) dot (1, 0, 3, 6, 10) = (5 + 0 + 9 + 12 + 10). (End)

Also the number of permutations of length n that can be sorted by a single block interchange (in the sense of Christie). - Vincent Vatter, Aug 21 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• D. A. Christie, Sorting Permutations by Block-Interchanges, Inf. Process. Lett. 60 (1996), 165-169.
• Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
• C. Homberger and V. Vatter, On the effective and automatic enumeration of polynomial permutation classes. [Broken link]
• C. Homberger, V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv preprint arXiv:1308.4946 [math.CO], 2013-2015.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

5th row of A145153. See row 5 of A145140/A145141 for rational coefficients and A145142 for 24 * coefficients of polynomial.

Programs

Maple

a:= n-> 1+ (6+ (11+ (6+ n) *n) *n) *n/24: seq(a(n), n=0..40);
# second Maple program:
with(combinat): seq(binomial(n+3, 4)+1, n=0..40); # Zerinvary Lajos, Mar 24 2009

Mathematica

a=b=s=0;lst={a};Do[a+=n;b+=a;s+=b;AppendTo[lst,s],{n,6!}];lst+1 (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)
CoefficientList[Series[(x^4 - 4 x^3 + 6 x^2 - 3 x + 1) / (1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 06 2013 *)

PARI

Vec((x^4-4*x^3+6*x^2-3*x+1)/(1-x)^5 + O(x^50)) \\ Altug Alkan, Nov 24 2015

Formula

G.f.: (x^4-4*x^3+6*x^2-3*x+1) / (1-x)^5.

a(n) = C(n+3,4)+1. - Zerinvary Lajos, Mar 24 2009