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 A207611 #26 Sep 19 2024 12:02:27
%S A207611 1,2,1,3,2,1,5,4,2,1,8,8,5,2,1,13,15,11,6,2,1,21,28,23,14,7,2,1,34,51,
%T A207611 47,32,17,8,2,1,55,92,93,70,42,20,9,2,1,89,164,181,148,97,53,23,10,2,
%U A207611 1,144,290,346,306,217,128,65,26,11,2,1,233,509,653,619,472
%N A207611 Triangle of coefficients of polynomials v(n,x) jointly generated with A207610; see Formula section.
%C A207611 Column 1: Fibonacci numbers, A000045
%C A207611 Column 2: A029907
%C A207611 Row sums: A003945.
%C A207611 For a discussion and guide to related arrays, see A208510.
%C A207611 Subtriangle of the triangle given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 25 2012
%F A207611 u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + x*v(n-1,x)+1, where u(1,x)=1, v(1,x)=1.
%F A207611 T(n,k) = T(n-1,k) + (n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(1,0) = T(2,1) = 1, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k >= n.
%e A207611 First five rows:
%e A207611 1;
%e A207611 2, 1;
%e A207611 3, 2, 1;
%e A207611 5, 4, 2, 1;
%e A207611 8, 8, 5, 2, 1;
%e A207611 From _Philippe Deléham_, Mar 25 2012: (Start)
%e A207611 (0, 2, -1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, ...) begins:
%e A207611 1;
%e A207611 0, 1;
%e A207611 0, 2, 1;
%e A207611 0, 3, 2, 1;
%e A207611 0, 5, 4, 2, 1;
%e A207611 0, 8, 8, 5, 2, 1;
%e A207611 0, 13, 15, 11, 6, 2, 1;
%e A207611 0, 21, 28, 23, 14, 7, 2, 1; (End)
%t A207611 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207611 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207611 v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1
%t A207611 Table[Factor[u[n, x]], {n, 1, z}]
%t A207611 Table[Factor[v[n, x]], {n, 1, z}]
%t A207611 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207611 TableForm[cu]
%t A207611 Flatten[%] (* A207610 *)
%t A207611 Table[Expand[v[n, x]], {n, 1, z}]
%t A207611 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207611 TableForm[cv]
%t A207611 Flatten[%] (* A207611 *)
%t A207611 T[ n_, k_] := Which[k<0 || n<0, 0, n<2, Boole[k<=n] + Boole[k==0&&n==1], True, T[n, k] = T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* _Michael Somos_, Sep 19 2024 *)
%o A207611 (Python)
%o A207611 from sympy import Poly
%o A207611 from sympy.abc import x
%o A207611 def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
%o A207611 def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
%o A207611 def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
%o A207611 for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017
%o A207611 (PARI) {T(n, k) = if(k<0 || n<0, 0, n<2, (k<=n) + (k==0 && n==1), T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* _Michael Somos_, Sep 19 2024 */
%Y A207611 Cf. A207610, A208510.
%K A207611 nonn,tabl
%O A207611 1,2
%A A207611 _Clark Kimberling_, Feb 19 2012