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A208513 Triangle of coefficients of polynomials u(n,x) jointly generated with A111125; see the Formula section.

%I A208513 #29 Feb 16 2025 08:33:16
%S A208513 1,1,1,1,4,1,1,9,6,1,1,16,20,8,1,1,25,50,35,10,1,1,36,105,112,54,12,1,
%T A208513 1,49,196,294,210,77,14,1,1,64,336,672,660,352,104,16,1,1,81,540,1386,
%U A208513 1782,1287,546,135,18,1,1,100,825,2640,4290,4004,2275,800,170,20,1
%N A208513 Triangle of coefficients of polynomials u(n,x) jointly generated with A111125; see the Formula section.
%C A208513 The columns of A208513 are identical to those of A208509.  Here, however, the alternating row sums are periodic (with period 1,0,-2,-3,-2,0).
%C A208513 From _Tom Copeland_, Nov 07 2015: (Start)
%C A208513 These polynomials may be expressed in terms of the Faber polynomials of A263916, similar to A127677.
%C A208513 Rephrasing notes in A111125: Append an initial column of zeros except for a 1 at the top to A111125. Then the rows of this entry contain the partial sums of the column sequences of modified A111125; therefore, the difference of consecutive pairs of rows of this entry, modified by appending an initial row of zeros to it, generates the modified A111125. (End)
%H A208513 G. C. Greubel, <a href="/A208513/b208513.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A208513 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce polynomials</a>
%F A208513 Coefficients of u(n, x) from the mixed recurrence relations:
%F A208513   u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A208513   v(n,x) = u(n-1,x) + (x+1)*v(n-1,x) + 1,
%F A208513 where u(1,x) = 1, u(2,x) = 1+x, v(1,x) = 1, v(2,x) = 3+x.
%F A208513 From _Peter Bala_, May 01 2012: (Start)
%F A208513 Working with an offset of 0: T(n,0) = 1; T(n,k) = (n/k)*binomial(n+k-1,2*k-1) = (n/k)*A078812(n,k) for k > 0. Cf. A156308.
%F A208513 O.g.f.: ((1-t)^2 + t^2*x)/((1-t)*((1-t)^2-t*x)) = 1 + (1+x)*t + (1+4*x+x^2)*t^2 + ....
%F A208513 u(n+1,x) = -1 + (b(2*n,x) + 1)/b(n,x), where b(n,x) = Sum_{k = 0..n} binomial(n+k, 2*k)*x^k are the Morgan-Voyce polynomials of A085478.
%F A208513 This triangle is formed from the even numbered rows of A211956 with a factor of 2^(k-1) removed from the k-th column entries.
%F A208513 (End)
%F A208513 T(n, k) = (2*(n-1)/(n+k-2))*binomial(n+k-2, 2*k-2). - _G. C. Greubel_, Feb 02 2022
%e A208513 First five rows:
%e A208513   1;
%e A208513   1,  1;
%e A208513   1,  4,  1;
%e A208513   1,  9,  6, 1;
%e A208513   1, 16, 20, 8, 1;
%e A208513 First five polynomials u(n,x):
%e A208513   u(1,x) = 1;
%e A208513   u(2,x) = 1 +    x;
%e A208513   u(3,x) = 1 +  4*x +    x^2;
%e A208513   u(4,x) = 1 +  9*x +  6*x^2 +   x^3;
%e A208513   u(5,x) = 1 + 16*x + 20*x^2 + 8*x^3 + x^4;
%t A208513 (* First program *)
%t A208513 u[1, x_]:=1; v[1, x_]:=1; z=16;
%t A208513 u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
%t A208513 v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x] + 1;
%t A208513 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208513 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208513 cu = Table[CoefficientList[u[n, x], x], {n,z}];
%t A208513 TableForm[cu]
%t A208513 Flatten[%]  (* A208513 *)
%t A208513 Table[Expand[v[n, x]], {n,z}]
%t A208513 cv = Table[CoefficientList[v[n, x], x], {n,z}];
%t A208513 TableForm[cv]
%t A208513 Flatten[%]  (* A111125 *)
%t A208513 (* Second program *)
%t A208513 T[n_, k_]:= If[k==1, 1, ((n-1)/(k-1))*Binomial[n+k-3, 2*k-3]];
%t A208513 Table[T[n, k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Feb 02 2022 *)
%o A208513 (Magma)
%o A208513 A208513:= func< n,k | k eq 1 select 1 else (2*(n-1)/(n+k-2))*Binomial(n+k-2, 2*k-2) >;
%o A208513 [A208513(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 02 2022
%o A208513 (Sage)
%o A208513 def A208513(n,k): return 1 if (k==1) else ((n-1)/(k-1))*binomial(n+k-3, 2*k-3)
%o A208513 flatten([[A208513(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 02 2022
%Y A208513 Cf. A085478, A078812, A111125, A156308, A208509, A211956.
%Y A208513 Cf. A127677, A263916.
%K A208513 nonn,tabl
%O A208513 1,5
%A A208513 _Clark Kimberling_, Feb 28 2012