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 A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)). 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 Cf. A003688, A003945. Sequence in context: A325013 A152685 A210287 * A089511 A246257 A210744 Adjacent sequences: A116409 A116410 A116411 * A116413 A116414 A116415 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Feb 13 2006 STATUS approved

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Last modified April 1 09:24 EDT 2023. Contains 361688 sequences. (Running on oeis4.)