The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A209139 Triangle of coefficients of polynomials u(n,x) jointly generated with A209140; see the Formula section. 3
 1, 2, 1, 3, 5, 3, 5, 12, 15, 7, 8, 27, 45, 42, 17, 13, 55, 119, 151, 116, 41, 21, 108, 282, 458, 480, 315, 99, 34, 205, 630, 1228, 1631, 1467, 845, 239, 55, 381, 1343, 3054, 4849, 5502, 4358, 2244, 577, 89, 696, 2769, 7173, 13218, 17895, 17838, 12666 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Column 1: A000045 (Fibonacci numbers). Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1... For a discussion and guide to related arrays, see A208510. Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012 LINKS FORMULA u(n,x) = u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x), where u(1,x)=1, v(1,x)=1. From Philippe Deléham, Apr 11 2012: (Start) As DELTA-triangle T(n,k) with 0 <= k <= n: G.f.: (1-2*y*x-y*x^2-y^2*x^2)/(1-x-x^2-2*y*x-y^2*x^2). T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End) EXAMPLE First five rows: 1; 2, 1; 3, 5, 3; 5, 12, 15, 7; 8, 27, 45, 42, 17; First three polynomials u(n,x): 1 2 + x 3 + 5x + 3x^2 From Philippe Deléham, Apr 11 2012: (Start) (1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins: 1; 1, 0; 2, 1, 0; 3, 5, 3, 0; 5, 12, 15, 7, 0; 8, 27, 45, 42, 17, 0; 13, 55, 119, 151, 116, 41, 0; (End) MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]; v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x]; 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[%] (* A209139 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A209140 *) CROSSREFS Cf. A209140, A208510. Sequence in context: A258236 A258244 A258248 * A257161 A253676 A182939 Adjacent sequences: A209136 A209137 A209138 * A209140 A209141 A209142 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 05 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 4 20:32 EST 2022. Contains 358570 sequences. (Running on oeis4.)