A105495 Triangle read by rows: T(n,k) is the number of compositions of n into k parts when parts equal to q are of q^2 kinds.

1, 4, 1, 9, 8, 1, 16, 34, 12, 1, 25, 104, 75, 16, 1, 36, 259, 328, 132, 20, 1, 49, 560, 1134, 752, 205, 24, 1, 64, 1092, 3312, 3338, 1440, 294, 28, 1, 81, 1968, 8514, 12336, 7815, 2456, 399, 32, 1, 100, 3333, 19800, 39572, 35004, 15765, 3864, 520, 36, 1, 121, 5368

Offset: 1

Author

Emeric Deutsch, Apr 10 2005

Keywords

• nonn
• tabl

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

Examples

			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
  ...
		

Crossrefs

Cf. A000290, A084938.

Row sums yield A033453.

Programs

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 *)

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

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)