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.
A219865 := proc(n) A219795(n+1)-A219795(n) ; end proc: # R. J. Mathar, Jan 06 2013
The coefficients and polynomials are 1; 1 1, 0; x 1, 0, 2; x^2 + 2 1, 0, 1, 0; x^3 + x 1, 0, 0, 0, -2; x^4 - 2 1, 0, -1, 0, -3, 0; x^5 - x^3 - 3*x 1, 0, -2, 0, -3, 0, 2; x^6 - 2*x^4 - 3*x^2 + 2 1, 0, -3, 0, -2, 0, 5, 0; x^7 - 3*x^5 - 2*x^3 + 5*x 1, 0, -4, 0, 0, 0, 8, 0, -2; x^8 - 4*x^6 + 8*x^2 - 2 1, 0, -5, 0, 3, 0, 10, 0, -7, 0; x^9 - 5*x^7 + 3*x^5 + 10*x^3 - 7*x
A053119:= func< n,k | (1/2)*(-1)^Floor(3*k/2)*(1+(-1)^k)*Binomial(n - Floor(k/2), n-k) >; A135929:= func< n,k | n eq 0 select 1 else A053119(n, k) + 3*A053119(n-2, k-2) >; [A135929(n,k): k in [0..n], n in [0..16]]; // G. C. Greubel, Apr 24 2023
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
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 *) (* Second program *) 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 *)
def A053119(n,k): return (-1)^(3*k/2)*((k+1)%2)*binomial(n-k/2, n-k) def A135929(n,k): return 1 if (n==0) else A053119(n, k) + 3*A053119(n-2, k-2) flatten([[A135929(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Apr 24 2023
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