OFFSET
1,2
COMMENTS
Triangle T(n,k)=
1. Riordan Array (1,(x+x^2)/(1-x)^3) without first column.
2. Riordan Array ((1+x)/(1-x)^3,(x+x^2)/(1-x)^3) numbering triangle (0,0).
[Vladimir Kruchinin, Nov 25 2011]
Triangle T(n,k), 1<=k<=n, given by (0, 4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 20 2012
T is the convolution triangle of the squares (A000290). - Peter Luschny, Oct 19 2022
FORMULA
G.f.: t*z*(1+z)/((1-z)^3-t*z*(1+z)).
From Vladimir Kruchinin, Nov 25 2011: (Start)
G.f.: ((x+x^2)/(1-x)^3)^k = Sum_{n>=k} T(n,k)*x^n.
T(n,k) = Sum{i=0..n-k} binomial(k,i)*binomial(n+2*k-i-1,3*k-1). (End)
EXAMPLE
T(3,2)=8 because we have (1,2),(1,2'),(1,2"),(1,2'"),(2,1),(2',1),(2",1) and (2'",1).
Triangle begins:
1;
4,1;
9,8,1;
16,34,12,1;
25,104,75,16,1;
...
Triangle (0, 4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins :
1
0, 1
0, 4, 1
0, 9, 8, 1
0, 16, 34, 12, 1
0, 25, 104, 75, 16, 1
...
MAPLE
G:=t*z*(1+z)/((1-z)^3-t*z*(1+z)): Gser:=simplify(series(G, z=0, 13)): for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 11 do seq(coeff(P[n], t^k), k=1..n) od; # yields sequence in triangular form
# Alternatively:
T := proc(k, n) option remember;
if k=n then 1 elif k=0 then 0 else add(i^2*T(k-1, n-i), i=1..n-k+1) fi end:
A105495 := (n, k) -> T(k, n):
for n from 1 to 9 do seq(A105495(n, k), k=1..n) od; # Peter Luschny, Mar 12 2016
# Uses function PMatrix from A357368. Adds column 1, 0, 0, 0, ... to the left.
PMatrix(10, n -> n^2); # Peter Luschny, Oct 19 2022
MATHEMATICA
nn=8; a=(x+x^2)/(1-x)^3; CoefficientList[Series[1/(1-y a), {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, Aug 31 2012 *)
PROG
(Maxima)
T(n, k):=sum(binomial(k, i)*binomial(n+2*k-i-1, 3*k-1), i, 0, n-k); /* Vladimir Kruchinin, Nov 25 2011 */
(SageMath)
@cached_function
def T(k, n):
if k==n: return 1
if k==0: return 0
return sum(i^2*T(k-1, n-i) for i in (1..n-k+1))
A105495 = lambda n, k: T(k, n)
for n in (0..6): print([A105495(n, k) for k in (0..n)]) # Peter Luschny, Mar 12 2016
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Apr 10 2005
STATUS
approved