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 A142470 A new {1,8,1} type symmetrical triangle sequence related to A008459: f(n,m)=Binomial[n, m]*Product[k!*(n + k)!/((m + k)!*(n - m + k)!), {k, 1, 2}]; t(n,m)=2^(m - n)*f(n, m)*Sum[Binomial[n, k]*Binomial[k, m], {k, m, n}]. 0
 1, 1, 1, 1, 8, 1, 1, 30, 30, 1, 1, 80, 300, 80, 1, 1, 175, 1750, 1750, 175, 1, 1, 336, 7350, 19600, 7350, 336, 1, 1, 588, 24696, 144060, 144060, 24696, 588, 1, 1, 960, 70560, 790272, 1728720, 790272, 70560, 960, 1, 1, 1485, 178200, 3492720, 14669424 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are: {1, 2, 10, 62, 462, 3852, 34974, 338690, 3452306, 36683660, 403472368}. From Peter Bala, May 08 2012: (Start) Define the action of the operator L on a sequence {a(i)}0<=i<=n by L{a(i)}0<=i<=n = {a(i)^2 - a(i-1)*a(i+1)}0<=i<=n with the conventions a(-1) = a(n+1) = 0. Extend the action of L to a lower triangular array T by letting L act on the rows of T. Then L acting on Pascal's triangle A007318 produces the triangle of Narayana numbers A001263 and L applied to A001263 produces the present triangle. Since the Narayana polynomials are real-rooted it follows by a theorem of Branden that the row polynomials of this array are also real-rooted. (End) REFERENCES P. Branden, Iterated sequences and the geometry of zeros, J. Reine Angew. Math. 658 (2011), 115-131 LINKS FORMULA A new {1,8,1} type symmetrical triangle sequence related to A008459: f(n,m)=Binomial[n, m]*Product[k!*(n + k)!/((m + k)!*(n - m + k)!), {k, 1, 2}]; t(n,m)=2^(m - n)*f(n, m)*Sum[Binomial[n, k]*Binomial[k, m], {k, m, n}]. From Peter Bala, May 08 2012: (Start) T(n,k) = C(n,k)^2 * product {i = 1..2} i!*(n+i)!/((k+i)!*(n-k+i)!) = C(n,k)*C(n+2,k)*C(n+2,k+1)*C(n+2,k+2)/(C(n+2,1)*C(n+2,2)). T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n,k)*C(n+1,k)*C(n+2,k+2) *C(n+3,k+1) = C(n,k)*A056939(n,k). (End) EXAMPLE {1}, {1, 1}, {1, 8, 1}, {1, 30, 30, 1}, {1, 80, 300, 80, 1}, {1, 175, 1750, 1750, 175, 1}, {1, 336, 7350, 19600, 7350, 336, 1}, {1, 588, 24696, 144060, 144060, 24696, 588, 1}, {1, 960, 70560, 790272, 1728720, 790272, 70560, 960, 1}, {1, 1485, 178200, 3492720, 14669424, 14669424, 3492720, 178200, 1485, 1}, {1, 2200, 408375, 13068000, 96049800, 184415616, 96049800, 13068000, 408375, 2200, 1} MATHEMATICA f[0, 0] = 1; f[n_, m_] := f[n, m] = Binomial[n, m]*Product[k!*(n + k)!/((m + k)!*(n - m + k)!), {k, 1, 2}]; t[n_, m_] = 2^(m - n)*f[n, m]*Sum[Binomial[n, k]*Binomial[k, m], {k, m, n}]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] CROSSREFS Cf. A008459. A001263, A056939. Sequence in context: A181543 A141696 A178122 * A168523 A144439 A157208 Adjacent sequences:  A142467 A142468 A142469 * A142471 A142472 A142473 KEYWORD nonn,uned AUTHOR Roger L. Bagula, Sep 20 2008 STATUS approved

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Last modified August 7 11:08 EDT 2020. Contains 336275 sequences. (Running on oeis4.)