# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a207608 Showing 1-1 of 1 %I A207608 #25 Apr 13 2020 06:24:12 %S A207608 1,2,3,3,4,11,3,5,26,20,3,6,50,74,29,3,7,85,204,149,38,3,8,133,469, %T A207608 547,251,47,3,9,196,952,1618,1160,380,56,3,10,276,1764,4110,4234,2124, %U A207608 536,65,3,11,375,3048,9318,13036,9262,3520,719,74,3,12,495,4983 %N A207608 Triangle of coefficients of polynomials u(n,x) jointly generated with A207609; see the Formula section. %C A207608 As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 03 2012 %F A207608 u(n,x) = u(n-1,x) + v(n-1,x), %F A207608 v(n,x) = 2x*u(n-1,x) + (x+1)v(n-1,x), %F A207608 where u(1,x)=1, v(1,x)=1. %F A207608 From _Philippe Deléham_, Mar 03 2012: (Start) %F A207608 As triangle T(n,k), 0 <= k <= n: %F A207608 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-1,k) + T(n-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n. %F A207608 G.f.: (1-y*x)/(1 - (2+y)*x - (y-1)*x^2). %F A207608 Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A025192(n), A001077(n), A180038(n) for x = 0, 1, 2, 3 respectively. (End) %e A207608 First five rows: %e A207608 1; %e A207608 2; %e A207608 3, 3; %e A207608 4, 11, 3; %e A207608 5, 26, 20, 3; %e A207608 Triangle (2, -1/2, 1/2, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, ...) begins: %e A207608 1; %e A207608 2, 0; %e A207608 3, 3, 0; %e A207608 4, 11, 3, 0; %e A207608 5, 26, 20, 3, 0; %e A207608 6, 50, 74, 29, 3, 0; %e A207608 7, 85, 204, 149, 38, 3, 0; %e A207608 ... - _Philippe Deléham_, Mar 03 2012 %t A207608 u[1, x_] := 1; v[1, x_] := 1; z = 16; %t A207608 u[n_, x_] := u[n - 1, x] + v[n - 1, x] %t A207608 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1) v[n - 1, x] %t A207608 Table[Factor[u[n, x]], {n, 1, z}] %t A207608 Table[Factor[v[n, x]], {n, 1, z}] %t A207608 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; %t A207608 TableForm[cu] %t A207608 Flatten[%] (* A207608 *) %t A207608 Table[Expand[v[n, x]], {n, 1, z}] %t A207608 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; %t A207608 TableForm[cv] %t A207608 Flatten[%] (* A207609 *) %o A207608 (Python) %o A207608 from sympy import Poly %o A207608 from sympy.abc import x %o A207608 def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x) %o A207608 def v(n, x): return 1 if n==1 else 2*x*u(n - 1, x) + (x + 1)*v(n - 1, x) %o A207608 def a(n): return Poly(u(n, x), x).all_coeffs()[::-1] %o A207608 for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017 %Y A207608 Cf. A207609. %K A207608 nonn,tabf %O A207608 1,2 %A A207608 _Clark Kimberling_, Feb 19 2012 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE