A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
1, 3, 1, 6, 6, 1, 12, 21, 9, 1, 24, 60, 45, 12, 1, 48, 156, 171, 78, 15, 1, 96, 384, 558, 372, 120, 18, 1, 192, 912, 1656, 1473, 690, 171, 21, 1, 384, 2112, 4608, 5160, 3225, 1152, 231, 24, 1, 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1, 1536, 10752
Offset: 0
Examples
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
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.
Programs
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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 *)
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
Comments