Real-rooted atlas

A178821 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\,n + n^{2} x\right)\,P_{n}(x) = \left(16 + 16\,x + 4\,x^{2} + 4\,n + 12\,n x + 5\,n x^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1101020
26012045225
32808406301401890
4112044805040224035013230
54032201603024020160630075681648
6134408064015120013440063000151201470459270
7422402956806652807392004620001663203234026402405700

All 15 rows

Technical details
Node id
n-3b2bfb7c3621f647 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e620d4a356abb37c

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:     10,      10;
  2:     60,     120,      45;
  3:    280,     840,     630,      140;
  4:   1120,    4480,    5040,     2240,      350;
  5:   4032,   20160,   30240,    20160,     6300,     756;
  6:  13440,   80640,  151200,   134400,    63000,   15120,    1470;
  7:  42240,  295680,  665280,   739200,   462000,  166320,   32340,   2640;
  8: 126720, 1013760, 2661120,  3548160,  2772000, 1330560,  388080,  63360,   4455;
  9: 366080, 3294720, 9884160, 15375360, 14414400, 8648640, 3363360, 823680, 115830, 7150;
%F Formula
The row polynomials satisfy (2*n+n^2*x)*P(n,x) = (4*n+16+(2*n^2+12*n+16)*x+(n^2+5*n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 16 + (2*n^2 + 12*n + 16)*x + (n^2 + 5*n + 4)*x^2)*P[n - 1, x])/(2*n + n^2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 16 + (2*n^2 + 12*n + 16)*x + (n^2 + 5*n + 4)*x^2)*v[i - 1])/(2*n + n^2*x)); 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

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. [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.