OFFSET
0,2
COMMENTS
Row sums are A003688. Diagonal sums are A116413. Product of A007318 and A116413 is A116414. Product of A007318 and A105475.
Subtriangle of triangle given by (0, 3, -1, 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 18 2012
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475
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
Number triangle T(n,k)=sum{j=0..n, C(k+1,j)*C(n-j,k)2^(n-k-j)}
From Vladimir Kruchinin, Mar 17 2011: (Start)
T((m+1)*n+r-1, m*n+r-1) * r/(m*n+r) = sum(k=1..n, k/n * T((m+1)*n-k-1, m*n-1) * T(r+k-1,r-1)), n>=m>1.
T(n-1,m-1) = m/n * sum(k=1..n-m+1, k*A003945(k-1)*T(n-k-1,m-2)), n>=m>1. (End)
G.f.: (1+x)/(1-(y+2)*x -y*x^2). - Philippe Deléham, Jan 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A104537(n), A110523(n), (-2)^floor(n/2), A057079(n), A003945(n), A003688(n+1), A123347(n), A180035(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 18 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(2,0) = T(2,1) = 6, T(2,2) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 31 2013
EXAMPLE
Triangle begins
1,
3, 1,
6, 6, 1,
12, 21, 9, 1,
24, 60, 45, 12, 1,
48, 156, 171, 78, 15, 1
Triangle T(n,k), 0<=k<=n, given by (0, 3, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins :
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 12, 21, 9, 1
0, 24, 60, 45, 12, 1
0, 48, 156, 171, 78, 15, 1
... - Philippe Deléham, Jan 18 2012
MATHEMATICA
With[{n = 10}, DeleteCases[#, 0] & /@ CoefficientList[Series[(1 + x)/(1 - (y + 2) x - y x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Feb 13 2006
STATUS
approved