0,2

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009: (Start)

Second right hand column (n-m=1) of the A156920 triangle.

The generating function of this sequence enabled the analysis of the polynomials A156921 and A156925.

(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 389

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

a(n)=sum{k=0..n+1, (n-k+1)3^k}=sum{k=0..n+1, k*3^(n-k+1)} - Paul Barry (pbarry(AT)wit.ie), Jul 30 2004

a(n)=sum{k=0..n, binomial(n+2, k+2)2^k} - Paul Barry (pbarry(AT)wit.ie), Jul 30 2004

a(-1)=0, a(0)=1, a(n)=4*a(n-1)-3*a(n-2)+1 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005

a(n) = right term of M^(n+1) * [1,0,0]; where M = the 3 X 3 matrix [1,0,0; 1,1,0; 1,1,3]. E.g. a(4) = 179 since M^5 = [1, 5, 179]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2006

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009: (Start)

a(n)=5*a(n-1)-7*a(n-2)+3*a(n-3) with a(0)=1, a(1)=5 and a(2)=18.

(End)

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-3*a[n-2]+1 od: seq(a[n], n=0..50); (Kristof)

A000340:=-1/(3*z-1)/(z-1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]

a:=n->sum(3^(n-j)*j, j=0..n): seq(a(n), n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2008

lst={}; s=0; Do[s+=(s+(n+=s)); AppendTo[lst, s], {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008]

(Other) sage: [(gaussian_binomial(n, 1, 3)-n)/2 for n in xrange(2, 27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009: (Start)

Cf. A156921, A156925, A156927, A156933. Other columns A156922, A156923, A156924.

Equals A156920 second right hand column.

Equals A142963 second right hand column divided by 2^n

Equals A156919 second right hand column divided by 2.

(End)

Sequence in context: A093374 A000745 A128553 this_sequence A034567 A133648 A099449

Adjacent sequences: A000337 A000338 A000339 this_sequence A000341 A000342 A000343

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)