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 A207607 Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section. 3
 1, 1, 2, 1, 5, 2, 1, 9, 9, 2, 1, 14, 25, 13, 2, 1, 20, 55, 49, 17, 2, 1, 27, 105, 140, 81, 21, 2, 1, 35, 182, 336, 285, 121, 25, 2, 1, 44, 294, 714, 825, 506, 169, 29, 2, 1, 54, 450, 1386, 2079, 1716, 819, 225, 33, 2, 1, 65, 660, 2508, 4719, 5005, 3185, 1240, 289, 37, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012 LINKS G. C. Greubel, Rows n = 1..100 of the triangle, flattened FORMULA u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = x*u(n-1,x) + (x+1)v(n-1,x), where u(1,x)=1, v(1,x)=1. T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Mar 03 2012 G.f.: (1-x+y*x)/(1-(y+2)*x+x^2). - Philippe Deléham, Mar 03 2012 For n >= 1, Sum{k=0..n} T(n,k)*x^k = A000012(n), A001906(n), A001834(n-1), A055271(n-1), A038761(n-1), A056914(n-1) for x = 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Mar 03 2012 T(n,k) = C(n+k-1,2*k) + 2*C(n+k-1,2*k-1). where C is binomial. - Yuchun Ji, May 23 2019 T(n,k) = T(n-1,k) + A207606(n,k-1). - Yuchun Ji, May 28 2019 Sum_{k=1..n} T(n, k)*x^k = { 4*(-1)^(n-1)*A016921(n-1) (x=-4), 3*(-1)^(n-1) * A130815(n-1) (x=-3), 2*(-1)^(n-1)*A010684(n-1) (x=-2), A057079(n+1) (x=-1), 0 (x=0), A001906(n) = Fibonacci(2*n) (x=1), 2*A001834(n-1) (x=2), 3*A055271(n-1) (x=3), 4*A038761(n-1) (x=4) }. - G. C. Greubel, Mar 15 2020 EXAMPLE First five rows:   1;   1,  2;   1,  5,  2;   1,  9,  9,  2;   1, 14, 25, 13,  2; Triangle (1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, ...) begins:   1;   1,  0;   1,  2,  0;   1,  5,  2,  0;   1,  9,  9,  2,  0;   1, 14, 25, 13,  2,  0;   1, 20, 55, 49, 17,  2,  0;   ... 1 = 2*1 - 1, 20 = 2*14 + 1 - 9, 55 = 2*25 + 14 - 9, 49 = 2*13 + 25 - 2, 17 = 2*2 + 1 - 0, 2 = 2*0 + 2 - 0. - Philippe Deléham, Mar 03 2012 MAPLE A207607:= (n, k) -> `if`(k=1, 1, binomial(n+k-3, 2*k-2) + 2*binomial(n+k-3, 2*k-3) ); seq(seq(A207607(n, k), k = 1..n), n = 1..10); # G. C. Greubel, Mar 15 2020 MATHEMATICA (* First program *) u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + v[n - 1, x] v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] Table[Factor[u[n, x]], {n, 1, z}] Table[Factor[v[n, x]], {n, 1, z}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%]  (* A207606 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]  (* A207607 *) (* Second program *) Table[If[k==1, 1, Binomial[n+k-3, 2*k-2] + 2*Binomial[n+k-3, 2*k-3]], {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Mar 15 2020 *) PROG (Python) from sympy import Poly from sympy.abc import x def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x) def v(n, x): return 1 if n==1 else x*u(n - 1, x) + (x + 1)*v(n - 1, x) def a(n): return Poly(v(n, x), x).all_coeffs()[::-1] for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017 (Sage) def T(n, k):     if k == 1: return 1     else: return binomial(n+k-3, 2*k-2) + 2*binomial(n+k-3, 2*k-3) [[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 15 2020 CROSSREFS Cf. A207606. Sequence in context: A259447 A228823 A249756 * A146024 A146023 A104766 Adjacent sequences:  A207604 A207605 A207606 * A207608 A207609 A207610 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Feb 19 2012 STATUS approved

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Last modified July 25 19:02 EDT 2021. Contains 346291 sequences. (Running on oeis4.)