A051744 a(n) = n*(n+1)*(n^2+5*n+18)/24.

2, 8, 21, 45, 85, 147, 238, 366, 540, 770, 1067, 1443, 1911, 2485, 3180, 4012, 4998, 6156, 7505, 9065, 10857, 12903, 15226, 17850, 20800, 24102, 27783, 31871, 36395, 41385, 46872, 52888, 59466, 66640, 74445, 82917, 92093, 102011, 112710, 124230, 136612

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999

Keywords

• nonn
• easy

Comments

a(n) is the number of binary words with length <= n+1 which contain at least one 0 and one 1 and have at most one ascent. - Amelia Gibbs, May 21 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Programs

Magma

I:=[2, 8, 21, 45, 85]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Apr 27 2012

Maple

A051744:=n->n*(n+1)*(n^2+5*n+18)/24: seq(A051744(n), n=1..50); # Wesley Ivan Hurt, Nov 01 2014

Mathematica

Table[1/24*n*(n+1)*(n^2+5*n+18),{n,60}] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
CoefficientList[Series[(2-2*x+x^2)/(1-x)^5,{x,0,50}],x] (* Vincenzo Librandi, Apr 27 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{2,8,21,45,85},50] (* Harvey P. Dale, Jan 02 2024 *)

PARI

a(n)=binomial(n+3,4)+binomial(n+1,2) \\ Charles R Greathouse IV, Mar 19 2012

Formula

a(n) = binomial(n+3, n-1) + binomial(n+1, n-1).

G.f.: x*(2-2*x+x^2)/(1-x)^5. - Colin Barker, Mar 19 2012

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Apr 27 2012

a(n) = sum_{k=1..n} sum{j=1..k} sum{i=1..j} (i + binomial(j,k)). - Wesley Ivan Hurt, Nov 01 2014

E.g.f.: (1/24)*x*(x^3+12*x^2+48*x+48)*exp(x). - Robert Israel, Nov 02 2014

a(n) = Sum_{i=1..n+1} Sum_{j=1...i-1} A077028(i,j). - Amelia Gibbs, May 21 2024