0,2

The array F(3;n,m) gives in the columns m>=1 the figurate numbers based on A016777, including the pentagonal numbers A000326, (see the W. Lang link).

This is the third member, d=3, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal (d=1), A029653 (d=2).

This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column nr. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n) is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(1+2*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(1+2*z)/(1-(1+x)*z).

The SW-NE diagonals give the Lucas numbers A000032: L(n)= sum( a(n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with L(0)=2. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Triangle T(n,k), read by rows, given by [3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.

Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhaeuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

W. Lang, First 10 rows and array of figurate numbers .

a(n, m)=F(3;n-m, m) for 0<= m <= n, else 0, with F(3;0, 0)=1, F(3;n, 0)=3 if n>=1 and F(3;n, m):=(3*n+m)*binomial(n+m-1, m-1)/m if m>=1.

G.f. column m (without leading zeros): (1+2*x)/(1-x)^(m+1), m>=0.

Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=3 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).

T(n, k) = C(n, k) + 2*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 28 2005

Equals M * A007318, where M = an infinite triangular matrix with all 1's in the main diagonal and all 2's in the subdiagonal. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 01 2007

Sum_{k, 0<=k<=n}T(n,k)=A151821(n+1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]

[1]; [3,1]; [3,4,1]; [3,7,5,1]; ...

Column sequences for m=1..9: A016777, A000326 (pentagonal), A002411, A001296, A051836, A051923, A050494, A053367, A053310.

Cf. A029653 (2, 1) Pascal triangle.

Cf. A093561(d=4).

Sequence in context: A107638 A104765 A064884 this_sequence A131504 A008311 A081772

Adjacent sequences: A093557 A093558 A093559 this_sequence A093561 A093562 A093563

nonn,tabl,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Apr 22 2004

Incorrect connection with A046055 deleted by N. J. A. Sloane, Jul 08 2009