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 A208513 Triangle of coefficients of polynomials u(n,x) jointly generated with A111125; see the Formula section. 9
 1, 1, 1, 1, 4, 1, 1, 9, 6, 1, 1, 16, 20, 8, 1, 1, 25, 50, 35, 10, 1, 1, 36, 105, 112, 54, 12, 1, 1, 49, 196, 294, 210, 77, 14, 1, 1, 64, 336, 672, 660, 352, 104, 16, 1, 1, 81, 540, 1386, 1782, 1287, 546, 135, 18, 1, 1, 100, 825, 2640, 4290, 4004, 2275, 800, 170 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The columns of A208513 are identical to those of A208509.  Here, however, the alternating row sums are periodic (with period 1,0,-2,-3,-2,0). From Tom Copeland, Nov 07 2015: (Start) These polynomials may be expressed in terms of the Faber polynomials of A263916, similar to A127677. Rephrasing notes in A111125: Append an initial column of zeros except for a 1 at the top to A111125. Then the rows of this entry contain the partial sums of the column sequences of modified A111125; therefore, the difference of consecutive pairs of rows of this entry, modified by appending an initial row of zeros to it, generates the modified A111125. (End) LINKS Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials FORMULA u(n,x)=u(n-1,x)+x*v(n-1,x), v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1, where u(1,x)=1, v(1,x)=1. From Peter Bala, May 01 2012: (Start) Working with an offset of 0: T(n,0) = 1; T(n,k) = n/k*binomial(n+k-1,2*k-1) = n/k*A078812(n,k) for k > 0. Cf. A156308. O.g.f.: ((1-t)^2+t^2*x)/((1-t)*((1-t)^2-t*x)) = 1 + (1+x)*t + (1+4*x+x^2)*t^2 + .... u(n+1,x) = -1 + (b(2*n,x)+1)/b(n,x), where b(n,x) := sum {k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478. This triangle is formed from the even numbered rows of A211956 with a factor of 2^(k-1) removed from the k-th column entries. (End) EXAMPLE First five rows: 1 1...1 1...4...1 1...9...6...1 1...16...20...8...1 First five polynomials u(n,x): 1 1 + x 1 + 4x + x^2 1 + 9x + 6x^2 + x^3 1 + 16x + 20x^2 + 8x^3 + x^4 MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%]  (* A208513 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]  (* A111125 *) CROSSREFS Cf. A111125, A208509. A078812, A085478, A156308, A211956. Cf. A263916, A127677. Sequence in context: A136100 A244811 A183153 * A141905 A114188 A110511 Adjacent sequences:  A208510 A208511 A208512 * A208514 A208515 A208516 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Feb 28 2012 STATUS approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)