Real-rooted atlas

A145324 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 8\,n x - 3\,n^{2} - 16\,n^{2} x - 16\,n^{2} x^{2} - n^{3} - 8\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,n + 8\,n x + 2\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131619
212136288436
36010644704614411972
4360878460144162304153600385192
525207876873723230607365944320442368014247256
620160771264920611252009216154680320235118592144506880596312544
718144082316161198756808608770563472491520805579161610078322688528482304027880594656

All 15 rows

Technical details
Node id
n-93aa65cf65b5d50b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7c9e990866ba240f

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        3,         16;
  2:       12,        136,         288;
  3:       60,       1064,        4704,         6144;
  4:      360,       8784,       60144,       162304,        153600;
  5:     2520,      78768,      737232,      3060736,       5944320,       4423680;
  6:    20160,     771264,     9206112,     52009216,     154680320,     235118592,      144506880;
  7:   181440,    8231616,   119875680,    860877056,    3472491520,    8055791616,    10078322688,     5284823040;
  8:  1814400,   95379840,  1643264064,  14362803200,   73464382720,  230569549824,   439126474752,   467499220992,   214035333120;
  9: 19958400, 1194330240, 23798695104, 245631585280, 1526248369920, 6062555418624, 15590303547392, 25216844562432, 23391853608960, 9512681472000;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(8*n^3+16*n^2+8*n)*x+(16*n^3+16*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(8*n^2+8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x + (16*n^3 + 16*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x + (16*n^3 + 16*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.