Real-rooted atlas

basis-map coefficients of A026729 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - 4\,x + 4\,n x\right)\,P_{n}(x) = \left(1 + 4\,x + 4\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2189
31164865
4124144256425
5132288102412802625
614048025606400614415625
7148720512019200368642867290625
81561008896044800129024200704131072515625
91641344143368960034406480281610485765898242890625
10172172821504161280774144240844847185925308416262144016015625
11180216030720268800154828860211201572864026542080262144001153433687890625
12188264042240422400283852813246464432537609732096014417920012687769650331648478515625
131963168563206336004866048264929281038090242919628805767168007612661766039797762181038082587890625
141104374473216915200790732849201152224919552759103488187432960032988200963925868544283534950493952409613916015625
151112436893184128128012300288861020164498391041771241472524812288011545870336183207198721984744652813153337344402653184074462890625
Technical details
Node id
n-20406d92a36f9204 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-14ef8ed0e797c5a3

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): yes

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,   48;
   4: 1, 24,  144,   256;
   5: 1, 32,  288,  1024,   1280;
   6: 1, 40,  480,  2560,   6400,   6144;
   7: 1, 48,  720,  5120,  19200,  36864,   28672;
   8: 1, 56, 1008,  8960,  44800, 129024,  200704,  131072;
   9: 1, 64, 1344, 14336,  89600, 344064,  802816, 1048576,  589824;
  10: 1, 72, 1728, 21504, 161280, 774144, 2408448, 4718592, 5308416, 2621440;
%F Formula
The row polynomials satisfy (1+(4*n-4)*x)*P(n,x) = (1+(4*n+4)*x+16*n*x^2)*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 + (4*n + 4)*x + 16*n*x^2)*P[n - 1, x])/(1 + (4*n - 4)*x)]];
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 + (4*n + 4)*x + 16*n*x^2)*v[i - 1])/(1 + (4*n - 4)*x)); 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.