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 A135929 #61 Apr 27 2023 17:04:43
%S A135929 1,1,0,1,0,2,1,0,1,0,1,0,0,0,-2,1,0,-1,0,-3,0,1,0,-2,0,-3,0,2,1,0,-3,
%T A135929 0,-2,0,5,0,1,0,-4,0,0,0,8,0,-2,1,0,-5,0,3,0,10,0,-7,0,1,0,-6,0,7,0,
%U A135929 10,0,-15,0,2,1,0,-7,0,12,0,7,0,-25,0,9,0,1,0,-8,0,18,0,0,0,-35,0,24,0,-2
%N A135929 Triangle read by rows: row n gives coefficients of polynomial P_n(x)= U_{n}(x,1) + 3 * U_{n-2}(x,1) where U is the Chebyshev polynomial of the second kind, in order of decreasing exponents.
%C A135929 Take a(0)=-2 instead of 1. The recurrence begins immediately (at the third instead of the fourth polynomial). Companion: A192011(n). - _Paul Curtz_, Sep 20 2011
%D A135929 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972; see Chapter 22.
%H A135929 Michael De Vlieger, <a href="/A135929/b135929.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A135929 Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, <a href="https://doi.org/10.3934/era.2020057">Recursive sequences and Girard-Waring identities with applications in sequence transformation</a>, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
%F A135929 G.f.: (1+3*t^2)/(1-x*t+t^2).
%F A135929 P_n(x) = U_{n}(x,1) + 3 * U_{n-2}(x,1) for n>=2. - _Max Alekseyev_, Dec 04 2009
%F A135929 P_n(x) = S_{n}(x) + 3*S_{n-2}(x), with Chebyshev Polynomials S_n(x) defined in A049310 and A053119. - _R. J. Mathar_, Dec 07 2009
%F A135929 P_0(x)=1, P_1(x)=x, P_2(x)=x^2+2, and P_n(x)= x*P_{n-1}(x) - P_{n-2}(x) for n>=3. - _Paul Curtz_, Aug 14 2011
%F A135929 From _G. C. Greubel_, Apr 24 2023: (Start)
%F A135929 T(n, k) = A053119(n, k) + 3*A053119(n-2, k-2), with T(0,0) = 1.
%F A135929 Sum_{k=0..n} T(n, k) = A138034(n). (End)
%e A135929 The coefficients and polynomials are
%e A135929 1; 1
%e A135929 1, 0; x
%e A135929 1, 0, 2; x^2 + 2
%e A135929 1, 0, 1, 0; x^3 + x
%e A135929 1, 0, 0, 0, -2; x^4 - 2
%e A135929 1, 0, -1, 0, -3, 0; x^5 - x^3 - 3*x
%e A135929 1, 0, -2, 0, -3, 0, 2; x^6 - 2*x^4 - 3*x^2 + 2
%e A135929 1, 0, -3, 0, -2, 0, 5, 0; x^7 - 3*x^5 - 2*x^3 + 5*x
%e A135929 1, 0, -4, 0, 0, 0, 8, 0, -2; x^8 - 4*x^6 + 8*x^2 - 2
%e A135929 1, 0, -5, 0, 3, 0, 10, 0, -7, 0; x^9 - 5*x^7 + 3*x^5 + 10*x^3 - 7*x
%p A135929 A135929 := proc(n, m) coeftayl( coeftayl( (1+3*t^2)/(1-x*t+t^2), t=0, n), x=0, n-m) ; end proc: seq(seq(A135929(n,m), m=0..n),n=0..14) ; # _R. J. Mathar_, Nov 03 2009
%t A135929 p[0, _]= 1; p[1, x_]:= x; p[2, x_]:= x^2+2; p[n_, x_]:= p[n, x] = x*p[n-1, x] - p[n-2, x]; row[n_]:= CoefficientList[p[n, x], x]; Table[row[n]//Reverse, {n, 0, 13}]//Flatten (* _Jean-François Alcover_, Nov 26 2012, after _Paul Curtz_'s formula *)
%t A135929 (* Second program *)
%t A135929 p=1; q=2; t[_, 0]=p; t[2, 2]=q; t[_, _?OddQ]=0; t[n_, k_] /; k > n = 0; t[n_ /; n >= 0, k_ /; k >= 0]:= t[n, k] = t[n-1, k] - t[n-2, k-2]; Table[t[n, k], {n, 0, 13}, {k, 0, n}]//Flatten (* _Jean-François Alcover_, Nov 27 2012, from recurrence *)
%o A135929 (Magma)
%o A135929 A053119:= func< n,k | (1/2)*(-1)^Floor(3*k/2)*(1+(-1)^k)*Binomial(n - Floor(k/2), n-k) >;
%o A135929 A135929:= func< n,k | n eq 0 select 1 else A053119(n, k) + 3*A053119(n-2, k-2) >;
%o A135929 [A135929(n,k): k in [0..n], n in [0..16]]; // _G. C. Greubel_, Apr 24 2023
%o A135929 (SageMath)
%o A135929 def A053119(n,k): return (-1)^(3*k/2)*((k+1)%2)*binomial(n-k/2, n-k)
%o A135929 def A135929(n,k): return 1 if (n==0) else A053119(n, k) + 3*A053119(n-2, k-2)
%o A135929 flatten([[A135929(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Apr 24 2023
%Y A135929 Cf. A049310, A053119, A138034, A135936, A137276 (row-reversed), A192011, A194084, A219795.
%K A135929 sign,tabl
%O A135929 0,6
%A A135929 _N. J. A. Sloane_, Mar 09 2008
%E A135929 Extended by _R. J. Mathar_, Nov 03 2009