A050407 - OEIS

%I #109 Jun 14 2024 08:01:23

%S 0,1,1,1,2,5,11,21,36,57,85,121,166,221,287,365,456,561,681,817,970,

%T 1141,1331,1541,1772,2025,2301,2601,2926,3277,3655,4061,4496,4961,

%U 5457,5985,6546,7141,7771,8437,9140,9881,10661,11481,12342,13245,14191

%N a(n) = n*(n^2 - 6*n + 11)/6.

%C Number of invertible shuffles of n-2 cards. - Adam C. McDougall (mcdougal(AT)stolaf.edu) and David Molnar (molnar(AT)stolaf.edu), Apr 09 2002

%C If Y is a 3-subset of an n-set X then, for n>=3, a(n-2) is the number of (n-3)-subsets of X which have neither one element nor two elements in common with Y. - _Milan Janjic_, Dec 28 2007

%C Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n+1)=-coeff(charpoly(A,x),x^(n-3)). - _Milan Janjic_, Jan 24 2010

%C Starting with offset 3: (1, 2, 5, 11, 21, ...) = triangle A144257 * [1,2,3,...]. - _Gary W. Adamson_, Feb 18 2010

%C (1 + 2x + 5x^2 + 11x^3 + ...) = (1 + 2x + 3x^2 + ...)*(1 + 2x^2 + 3x^3 + ...). - _Gary W. Adamson_, Jul 26 2010

%C Starting (1, 2, 5, 11, ...) = binomial transform of [1, 1, 2, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 25 2010

%C For n > 1: abs(abs(a(n+2) - a(n+1)) - abs(a(n+1) - a(n))) = n - 1; see also A086283. - _Reinhard Zumkeller_, Oct 17 2014

%C For n > 0, a(n) is the number of valid hook configurations of permutations of [n-1] that avoid the patterns 132 and 321. - _Colin Defant_, Apr 28 2019

%C For n >= 1, a(n+2) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 4 with exactly one descent. - _Jessica A. Tomasko_, Nov 15 2022

%C The number of bigrassmannian permutations in S_n is a(n+2) = binomial(n+1, 3) + 1. - _Joshua Swanson_, Jan 08 2024

%H Vincenzo Librandi, <a href="/A050407/b050407.txt">Table of n, a(n) for n = 0..1000</a>

%H Kassie Archer and Aaron Geary, <a href="https://arxiv.org/abs/2406.09369">Descents in powers of permutations</a>, arXiv:2406.09369 [math.CO], 2024.

%H Colin Defant, <a href="http://arxiv.org/abs/1904.10451">Motzkin intervals and valid hook configurations</a>, arXiv preprint arXiv:1904.10451 [math.CO], 2019.

%H Robert DiSario, <a href="http://www.jstor.org/stable/2695364">Problem 10931</a>, Amer. Math. Monthly, 109 (No. 3, 2002), 298.

%H J. B. Gil and J. Tomasko, <a href="https://doi.org/10.54550/ECA2022V2S4PP6">Restricted Grassmannian permutations</a>, ECA 2:4 (2022) Article S4PP6.

%H Nurul Hilda Syani Putri, Mashadi, and Sri Gemawati, <a href="https://doi.org/10.12988/imf.2018.815">Sequences from heptagonal pyramid corners of integer</a>, International Mathematical Forum, Vol. 13, 2018, no. 4, 193-200.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

%H Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, <a href="http://www.ece.nus.edu.sg/stfpage/eleak/pdf/akumar_todaes_2012.pdf">Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs</a>, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 25 2012

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F From _Paul Barry_, Jul 21 2003: (Start)

%F Diagonal sums of square array A086460 (starting 1, 1, 2, ...).

%F a(n+2) = 1 + n*(n+1)*(n-1)/6 = Sum_{k=0..n} (0^k + (n-k)*k). (End)

%F a(n) = binomial(n-1,3) + binomial(n-1,0), n>=0. - _Zerinvary Lajos_, Jul 24 2006

%F G.f.: x*(1-3*x+3*x^2)/(1-x)^4. - _Colin Barker_, May 06 2012

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 22 2012

%F a(n) = A000292(n-3) + 1, n > 2. - _Ivan N. Ianakiev_, Apr 27 2014

%F E.g.f.: x*(6 - 3*x + x^2)*exp(x)/6. - _G. C. Greubel_, Oct 30 2019

%F a(n+2) = 1 + Sum_{i=3..4} binomial(n, i-1) for n >= 1. - _Jessica A. Tomasko_, Nov 15 2022

%p seq(binomial(n-1, 3) + 1, n = 0..46); # _Zerinvary Lajos_, Jul 24 2006

%t Table[n*(n^2-6*n+11)/6, {n,0,50}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 17 2008 *)

%t LinearRecurrence[{4,-6,4,-1},{0,1,1,1},50] (* _Vincenzo Librandi_, Jun 22 2012 *)

%t Join[{0,1,1},Nest[Accumulate,Range[0,50],2]+1] (* _Harvey P. Dale_, Sep 23 2017 *)

%o (Magma) I:=[0, 1, 1, 1]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jun 22 2012

%o (Haskell)

%o a050407 n = n * (n ^ 2 - 6 * n + 11) `div` 6

%o -- _Reinhard Zumkeller_, Oct 17 2014

%o (PARI) a(n)=n*(n^2-6*n+11)/6 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Python) for n in range(0,50): print(n*(n**2 - 6*n + 11)/6, end=', ') # _Stefano Spezia_, Jan 05 2019

%o (Sage) [n*(n^2-6*n+11)/6 for n in (0..50)] # _G. C. Greubel_, Oct 30 2019

%o (GAP) List([0..50], n-> n*(n^2-6*n+11)/6); # _G. C. Greubel_, Oct 30 2019

%Y Apart from initial terms, one more than the tetrahedral numbers A000292.

%Y Cf. A144257, A086283.

%K easy,nonn

%O 0,5

%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999