Real-rooted atlas

basis-map coefficients of A026729 → basis A013609 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(1 - 8\,x + 4\,x^{2} + 6\,n x - 8\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 - 4\,x^{2} + 6\,n x + 12\,n x^{2} + 4\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2189
31163653
4124108128261
51322165124001161
61403601280200011524833
7148540256060006912313619197
81567564480140002419221952819273629

All 15 rows

Technical details
Node id
n-8524bd9002d643c2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  8;
   3: 1, 16,   36;
   4: 1, 24,  108,   128;
   5: 1, 32,  216,   512,   400;
   6: 1, 40,  360,  1280,  2000,   1152;
   7: 1, 48,  540,  2560,  6000,   6912,   3136;
   8: 1, 56,  756,  4480, 14000,  24192,  21952,   8192;
   9: 1, 64, 1008,  7168, 28000,  64512,  87808,  65536,  20736;
  10: 1, 72, 1296, 10752, 50400, 145152, 263424, 294912, 186624, 51200;
%F Formula
The row polynomials satisfy (1+(6*n-8)*x+(4*n^2-8*n+4)*x^2)*P(n,x) = (1+6*n*x+(4*n^2+12*n-4)*x^2+8*n^2*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 6*n*x + (4*n^2 + 12*n - 4)*x^2 + 8*n^2*x^3)*P[n - 1, x])/(1 + (6*n - 8)*x + (4*n^2 - 8*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + 6*n*x + (4*n^2 + 12*n - 4)*x^2 + 8*n^2*x^3)*v[i - 1])/(1 + (6*n - 8)*x + (4*n^2 - 8*n + 4)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.