OFFSET
1,2
COMMENTS
The column of index 0 contains a 1 followed by zeros and is not reproduced in this triangle.
Triangle T(n,k), 1 <= k <= n, given by (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 26 2012
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
FORMULA
T(n,m) = Sum_{k=m..n} binomial(n-1,k-1) * Sum_{i=ceiling((k-m)/2)..k-m} binomial(i,k-m-i)*binomial(m+i-1,m-1), 0<m<=n.
T(n,m) = Sum_{i=1..n-m+1} A001519(i)*T(n-i,m-1).
T(n,1) = A001519(n).
Sum_{m=1..n} T(n,m) = A007052(n-1).
G.f.: (1-3x+x^2)/(1-(3+y)*x + (1+y)*x^2). - Philippe Deléham, Jan 26 2012
EXAMPLE
Triangle begins:
1;
2, 1;
5, 4, 1;
13, 14, 6, 1;
34, 46, 27, 8, 1;
89, 145, 107, 44, 10, 1;
From Philippe Deléham, Jan 26 2012: (Start)
Triangle (0,2,1/2,1/2,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,...) begins:
1;
0, 1;
0, 2, 1;
0, 5, 4, 1;
0, 13, 14, 6, 1;
0, 34, 46, 27, 8, 1;
0, 89, 145, 107, 44, 10, 1; (End)
MAPLE
A188137 := proc(n, m) add( binomial(n-1, k-1) *add(binomial(i, k-m-i) *binomial(m+i-1, m-1), i=ceil((k-m)/2)..k-m), k=m..n) ; end proc:
seq(seq(A188137(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Mar 30 2011
MATHEMATICA
t[n_, m_] := Sum[ Binomial[n - 1, k - 1]*Sum[ Binomial[i, k - m - i]*Binomial[m + i - 1, m - 1], {i, Ceiling[(k - m)/2], k - m}], {k, m, n}]; Table[t[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013, translated from Maxima *)
PROG
(Maxima)
T(n, m):=sum(binomial(n-1, k-1) *sum(binomial(i, k-m-i) *binomial(m+i-1, m-1), i, ceiling((k-m)/2), k-m), k, m, n);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Mar 21 2011
STATUS
approved