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 A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0. 8
 1, 1, 2, 1, 3, 4, 1, 4, 8, 8, 1, 5, 13, 20, 16, 1, 6, 19, 38, 48, 32, 1, 7, 26, 63, 104, 112, 64, 1, 8, 34, 96, 192, 272, 256, 128, 1, 9, 43, 138, 321, 552, 688, 576, 256, 1, 10, 53, 190, 501, 1002, 1520, 1696, 1280, 512, 1, 11, 64, 253, 743, 1683, 2972, 4048 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Previous name was: Triangle of coefficients of polynomials v(n,x) jointly generated with A160232; see the Formula section. Row sums: (1,3,8,...), even-indexed Fibonacci numbers. Alt. row sums: (1,-1,2,-3,...), signed Fibonacci numbers. v(n,2) = A107839(n), v(n,n) = 2^(n-1), v(n+1,n) = A001792(n), v(n+2,n) = A049611, v(n+3,n) = A049612. Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012 Essentially triangle in A049600. - Philippe Deléham, Mar 23 2012 LINKS Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened FORMULA u(n,x) = u(n-1,x)+x*v(n-1,x), v(n,x) = u(n-1,x)+2x*v(n-1,x), where u(1,x) = 1, v(1,x) = 1. As DELTA-triangle with 0<=k<=n : T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 12 2012 G.f.: (1-2*y*x+y*x^2)/(1-x-2*y*x+y*x^2). - Philippe Deléham, Mar 12 2012 T(n,k) = A106195(n-1,n-k), k = 1..n. - Reinhard Zumkeller, Dec 16 2013 From Peter Bala, Aug 11 2015: (Start) The following remarks assume the row and column indexing start at 0. T(n,k) = Sum_{i = 0..k} 2^(k-i)*binomial(n-k,i)*binomial(k,i) = Sum_{i = 0..k} binomial(n-k+i,i)*binomial(k,i). Riordan array (1/(1 - x), x*(2 - x)/(1 - x)). O.g.f. 1/(1 - (2*t + 1)*x + t*x^2) = 1 + (1 + 2*t)*x + (1 + 3*t + 4*t^2)*x^2 + .... Read as a square array, this equals P * transpose(P^2), where P denotes Pascal's triangle A007318. (End) EXAMPLE First five rows:   1   1...2   1...3...4   1...4...8....8   1...5...13...20...16 First five polynomials v(n,x):   1   1 + 2x   1 + 3x + 4x^2   1 + 4x + 8x^2 + 8x^3   1 + 5x + 13x^2 + 20x^3 + 16x^4 (1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins :   1   1, 0   1, 2, 0   1, 3, 4, 0   1, 4, 8, 8, 0   1, 5, 13, 20, 16, 0   1, 6, 19, 38, 48, 32, 0 Triangle in A049600 begins :   0   0, 1   0, 1, 2   0, 1, 3, 4   0, 1, 4, 8, 8   0, 1, 5, 13, 20, 16   0, 1, 6, 19, 38, 48, 32 . Philippe Deléham, Mar 23 2012 MAPLE T := (n, k) -> hypergeom([n-k+1, -k], [1], -1): seq(lprint(seq(simplify(T(n, k)), k=0..n)), n=0..7); # Peter Luschny, May 20 2015 MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 13; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; v[n_, x_] := 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[%]   (* A160232 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]   (* A208341 *) PROG (Haskell) a208341 n k = a208341_tabl !! (n-1) !! (k-1) a208341_row n = a208341_tabl !! (n-1) a208341_tabl = map reverse a106195_tabl -- Reinhard Zumkeller, Dec 16 2013 (PARI) T(n, k) = sum(i = 0, k, 2^(k-i)*binomial(n-k, i)*binomial(k, i)); tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print(); ); \\ Michel Marcus, Aug 14 2015 CROSSREFS Cf. A160232, A000045, A049600, A106195. Sequence in context: A181851 A210231 A180378 * A201634 A210211 A283054 Adjacent sequences:  A208338 A208339 A208340 * A208342 A208343 A208344 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Feb 25 2012 EXTENSIONS New name from Peter Luschny, May 20 2015 Offset corrected, Joerg Arndt, Aug 12 2015 STATUS approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)