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 A211957 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478. 4
 1, 1, 1, 1, 4, 2, 1, 9, 12, 4, 1, 16, 40, 32, 8, 1, 25, 100, 140, 80, 16, 1, 36, 210, 448, 432, 192, 32, 1, 49, 392, 1176, 1680, 1232, 448, 64, 1, 64, 672, 2688, 5280, 5632, 3328, 1024, 128, 1, 81, 1080, 5544, 14256, 20592, 17472, 8640, 2304, 256, 1, 100, 1650, 10560, 34320, 64064, 72800, 51200, 21760, 5120, 512 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Triangle formed from the even numbered rows of A211956. The coefficients of the  Morgan-Voyce polynomials b(n,x) := sum {k = 0..n} binomial(n+k,2*k)*x^k are listed in A085478. The rational functions 1/2*(b(2*n,2*x) + 1)/b(n,2*x) turn out to be integer polynomials. Their coefficients are listed in this triangle. These polynomials occur as factors of the row polynomials R(n,x) of A211955. This triangle appears to be the row reverse of the unsigned triangle |A204021|. LINKS Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials FORMULA T(n,0) = 1 and for k > 0, T(n,k) = n/k*2^(k-1)*binomial(n+k-1,2*k-1) = 2^(k-1)*A208513(n,k). O.g.f.: ((1-t)-t*x)/((1-t)^2-2*t*x) = 1 + (1+x)*t + (1+4*x+2*x^2)*t^2 + .... n-th row polynomial R(n,x) = 1/2*(b(2*n,2*x) + 1)/b(n,2*x) = T(2*n,u), where u = sqrt((x+2)/2) and T(n,u) denotes the Chebyshev polynomial of the first kind. T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 16 2013 EXAMPLE Triangle begins .n\k.|..0....1....2....3....4....5....6....7 = = = = = = = = = = = = = = = = = = = = = = = ..0..|..1 ..1..|..1....1 ..2..|..1....4....2 ..3..|..1....9...12....4 ..4..|..1...16...40...32....8 ..5..|..1...25..100..140...80...16 ..6..|..1...36..210..448..432..192...32 ..7..|..1...49..392.1176.1680.1232..448...64 CROSSREFS A085478, A111125, A204021, A211955, A211956. Sequence in context: A160905 A208612 A183157 * A338397 A063983 A259985 Adjacent sequences:  A211954 A211955 A211956 * A211958 A211959 A211960 KEYWORD nonn,easy,tabl AUTHOR Peter Bala, Apr 30 2012 STATUS approved

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Last modified June 21 02:05 EDT 2021. Contains 345342 sequences. (Running on oeis4.)