[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A105495 revision #34

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.
2
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
1,2
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
Row sums yield A033453.
Sequence in context: A299615 A353770 A049762 * A256831 A010644 A137615
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Apr 10 2005
STATUS
approved