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A207607 Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.

%I A207607 #38 Apr 13 2020 05:56:58
%S A207607 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,
%T A207607 2,1,35,182,336,285,121,25,2,1,44,294,714,825,506,169,29,2,1,54,450,
%U A207607 1386,2079,1716,819,225,33,2,1,65,660,2508,4719,5005,3185,1240,289,37,2
%N A207607 Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
%C A207607 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
%H A207607 G. C. Greubel, <a href="/A207607/b207607.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A207607 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.
%F A207607 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - _Philippe Deléham_, Mar 03 2012
%F A207607 G.f.: (1-x+y*x)/(1-(y+2)*x+x^2). - _Philippe Deléham_, Mar 03 2012
%F A207607 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
%F A207607 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
%F A207607 T(n,k) = T(n-1,k) + A207606(n,k-1). - _Yuchun Ji_, May 28 2019
%F A207607 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
%e A207607 First five rows:
%e A207607   1;
%e A207607   1,  2;
%e A207607   1,  5,  2;
%e A207607   1,  9,  9,  2;
%e A207607   1, 14, 25, 13,  2;
%e A207607 Triangle (1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, ...) begins:
%e A207607   1;
%e A207607   1,  0;
%e A207607   1,  2,  0;
%e A207607   1,  5,  2,  0;
%e A207607   1,  9,  9,  2,  0;
%e A207607   1, 14, 25, 13,  2,  0;
%e A207607   1, 20, 55, 49, 17,  2,  0;
%e A207607   ...
%e A207607 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
%p A207607 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
%t A207607 (* First program *)
%t A207607 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207607 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207607 v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x]
%t A207607 Table[Factor[u[n, x]], {n, 1, z}]
%t A207607 Table[Factor[v[n, x]], {n, 1, z}]
%t A207607 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207607 TableForm[cu]
%t A207607 Flatten[%]  (* A207606 *)
%t A207607 Table[Expand[v[n, x]], {n, 1, z}]
%t A207607 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207607 TableForm[cv]
%t A207607 Flatten[%]  (* A207607 *)
%t A207607 (* Second program *)
%t A207607 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 *)
%o A207607 (Python)
%o A207607 from sympy import Poly
%o A207607 from sympy.abc import x
%o A207607 def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
%o A207607 def v(n, x): return 1 if n==1 else x*u(n - 1, x) + (x + 1)*v(n - 1, x)
%o A207607 def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
%o A207607 for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017
%o A207607 (Sage)
%o A207607 def T(n, k):
%o A207607     if k == 1: return 1
%o A207607     else: return binomial(n+k-3, 2*k-2) + 2*binomial(n+k-3, 2*k-3)
%o A207607 [[T(n, k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Mar 15 2020
%Y A207607 Cf. A207606.
%K A207607 nonn,tabl
%O A207607 1,3
%A A207607 _Clark Kimberling_, Feb 19 2012