This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A207606 #50 Apr 10 2020 06:16:59
%S A207606 1,2,3,2,4,7,2,5,16,11,2,6,30,36,15,2,7,50,91,64,19,2,8,77,196,204,
%T A207606 100,23,2,9,112,378,540,385,144,27,2,10,156,672,1254,1210,650,196,31,
%U A207606 2,11,210,1122,2640,3289,2366,1015,256,35,2,12,275,1782,5148,8008
%N A207606 Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
%C A207606 As triangle T(n,k) with 0 <= k <= n, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 03 2012
%H A207606 G. C. Greubel, <a href="/A207606/b207606.txt">Rows n = 1..101 of the triangle, flattened</a>
%F A207606 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 A207606 As triangle T(n,k) with 0 <= k <= n: g.f.: (1-y*x)/(1-(2+y)*x+x^2). - _Philippe Deléham_, Mar 03 2012
%F A207606 As triangle T(n,k) with 0 <= k <= n: Sum_{k=0..n} T(n,k)*x^k = A132677(n), A000034(n)*A057077(n), A057079(n), A000027(n+1), A001519(n+1), A001075(n), A002310(n), A038725(n), A172968(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively. - _Philippe Deléham_, Mar 03 2012
%F A207606 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - _Philippe Deléham_, Mar 03 2012
%F A207606 T(n,k) = C(n+k-1,2*k+1) + 2*C(n+k-1,2*k), where C is binomial. - _Yuchun Ji_, May 23 2019
%F A207606 T(n,k) = T(n-1,k) + A207607(n-1,k). - _Yuchun Ji_, May 28 2019
%e A207606 First five rows:
%e A207606 1;
%e A207606 2;
%e A207606 3, 2;
%e A207606 4, 7, 2;
%e A207606 5, 16, 11, 2;
%e A207606 Triangle (2, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, ...), 0 <= k <= n, begins:
%e A207606 1;
%e A207606 2, 0;
%e A207606 3, 2, 0;
%e A207606 4, 7, 2, 0;
%e A207606 5, 16, 11, 2, 0;
%e A207606 6, 30, 36, 15, 2, 0;
%e A207606 7, 50, 91, 64, 19, 2, 0;
%e A207606 8, 77, 196, 204, 100, 23, 2, 0;
%p A207606 T:= proc(n, k) option remember;
%p A207606 if k<0 or k>n then 0
%p A207606 elif k=0 then n+2
%p A207606 elif k=n then 2
%p A207606 else 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k)
%p A207606 fi; end:
%p A207606 1, seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Mar 15 2020
%t A207606 (* First program *)
%t A207606 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207606 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207606 v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x]
%t A207606 Table[Factor[u[n, x]], {n, 1, z}]
%t A207606 Table[Factor[v[n, x]], {n, 1, z}]
%t A207606 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207606 TableForm[cu]
%t A207606 Flatten[%] (* A207606 *)
%t A207606 Table[Expand[v[n, x]], {n, 1, z}]
%t A207606 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207606 TableForm[cv]
%t A207606 Flatten[%] (* A207607 *)
%t A207606 (* Second program *)
%t A207606 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, n+2, If[k==n, 2, 2*T[n-1, k] - T[n-2, k] + T[n-1, k-1] ]]]; Join[{1}, Table[T[n, k], {n, 0, 10}, {k, 0, n}]]//Flatten (* _G. C. Greubel_, Mar 15 2020 *)
%o A207606 (Python)
%o A207606 from sympy import Poly
%o A207606 from sympy.abc import x
%o A207606 def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
%o A207606 def v(n, x): return 1 if n==1 else x*u(n - 1, x) + (x + 1)*v(n - 1, x)
%o A207606 def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
%o A207606 for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017
%o A207606 (Sage)
%o A207606 @CachedFunction
%o A207606 def T(n, k):
%o A207606 if (k<0 or k>n): return 0
%o A207606 elif (k==1): return n+1
%o A207606 elif (k==n): return 2
%o A207606 else: return 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k)
%o A207606 [1]+[[T(n, k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Mar 15 2020
%Y A207606 Cf. A207607.
%K A207606 nonn,tabf
%O A207606 1,2
%A A207606 _Clark Kimberling_, Feb 19 2012