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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.

%I A105495 #34 Jun 03 2024 12:25:50
%S A105495 1,4,1,9,8,1,16,34,12,1,25,104,75,16,1,36,259,328,132,20,1,49,560,
%T A105495 1134,752,205,24,1,64,1092,3312,3338,1440,294,28,1,81,1968,8514,12336,
%U A105495 7815,2456,399,32,1,100,3333,19800,39572,35004,15765,3864,520,36,1,121,5368
%N 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.
%C A105495 Triangle T(n,k)=
%C A105495 1. Riordan Array (1,(x+x^2)/(1-x)^3) without first column.
%C A105495 2. Riordan Array ((1+x)/(1-x)^3,(x+x^2)/(1-x)^3) numbering triangle (0,0).
%C A105495 [_Vladimir Kruchinin_, Nov 25 2011]
%C A105495 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
%C A105495 T is the convolution triangle of the squares (A000290). - _Peter Luschny_, Oct 19 2022
%F A105495 G.f.: t*z*(1+z)/((1-z)^3-t*z*(1+z)).
%F A105495 From _Vladimir Kruchinin_, Nov 25 2011: (Start)
%F A105495 G.f.: ((x+x^2)/(1-x)^3)^k = Sum_{n>=k} T(n,k)*x^n.
%F A105495 T(n,k) = Sum{i=0..n-k} binomial(k,i)*binomial(n+2*k-i-1,3*k-1). (End)
%e A105495 T(3,2)=8 because we have (1,2),(1,2'),(1,2"),(1,2'"),(2,1),(2',1),(2",1) and (2'",1).
%e A105495 Triangle begins:
%e A105495   1;
%e A105495   4,1;
%e A105495   9,8,1;
%e A105495   16,34,12,1;
%e A105495   25,104,75,16,1;
%e A105495   ...
%e A105495 Triangle (0, 4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins :
%e A105495   1
%e A105495   0, 1
%e A105495   0, 4, 1
%e A105495   0, 9, 8, 1
%e A105495   0, 16, 34, 12, 1
%e A105495   0, 25, 104, 75, 16, 1
%e A105495   ...
%p A105495 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
%p A105495 # Alternatively:
%p A105495 T := proc(k,n) option remember;
%p A105495 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:
%p A105495 A105495 := (n,k) -> T(k,n):
%p A105495 for n from 1 to 9 do seq(A105495(n,k), k=1..n) od; # _Peter Luschny_, Mar 12 2016
%p A105495 # Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
%p A105495 PMatrix(10, n -> n^2); # _Peter Luschny_, Oct 19 2022
%t A105495 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 *)
%o A105495 (Maxima)
%o A105495 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 */
%o A105495 (SageMath)
%o A105495 @cached_function
%o A105495 def T(k,n):
%o A105495     if k==n: return 1
%o A105495     if k==0: return 0
%o A105495     return sum(i^2*T(k-1,n-i) for i in (1..n-k+1))
%o A105495 A105495 = lambda n,k: T(k, n)
%o A105495 for n in (0..6): print([A105495(n, k) for k in (0..n)]) # _Peter Luschny_, Mar 12 2016
%Y A105495 Cf. A000290, A084938.
%Y A105495 Row sums yield A033453.
%K A105495 nonn,tabl
%O A105495 1,2
%A A105495 _Emeric Deutsch_, Apr 10 2005