0,2

The array F(4;n,m) gives in the columns m>=1 the figurate numbers based on A016813, including the hexangonal numbers A000384, (see the W. Lang link).

This is the fourth member, d=4, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653 and A093560, for d=1,..,3.

This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). 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)=(1+3*z)/(1-(1+x)*z).

The SW-NE diagonals give A000285(n-1) = sum( a(n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with n=0 value 3. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs.

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(4;n-m, m) for 0<= m <= n, else 0, with F(4;0, 0)=1, F(4;n, 0)=4 if n>=1 and F(4;n, m):=(4*n+m)*binomial(n+m-1, m-1)/m if m>=1.

Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=4 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).

G.f. row m (without leading zeros): (1+3*x)/(1-x)^(m+1), m>=0.

T(n, k) = C(n, k) + 3*C(n-1, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 28 2005

[1]; [4,1]; [4,5,1]; [4,9,6,1]; ...

Row sums: A020714(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 3 for n=2 and 0 else.

Columns m=1..9: A016813, A000384 (hexagonal), A002412, A002417, A034263, A051947, A050483, A052181, A055843.

Cf. A093562 (d=5).

Sequence in context: A035646 A144034 A151783 this_sequence A081773 A167431 A110361

Adjacent sequences: A093558 A093559 A093560 this_sequence A093562 A093563 A093564

nonn,easy,tabl

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