Real-rooted atlas

A178822 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n + 2\,n^{2} x\right)\,P_{n}(x) = \left(10 + 40\,x + 40\,x^{2} + 2\,n + 28\,n x + 48\,n x^{2} + 4\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1124860
28467210081764
34485376161281433636288
4201632256145152258048161280598752
580641612809676802580480322560015482888491392
629568709632532224018923520354816003406233613246464107775360
71013762838528255467521135411202838528004087480323179151361038090241256352768

All 15 rows

Technical details
Node id
n-99d2da29c60c7f69 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7cabb77b45806852

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:      12,       48;
  2:      84,      672,      1008;
  3:     448,     5376,     16128,      14336;
  4:    2016,    32256,    145152,     258048,      161280;
  5:    8064,   161280,    967680,    2580480,     3225600,     1548288;
  6:   29568,   709632,   5322240,   18923520,    35481600,    34062336,    13246464;
  7:  101376,  2838528,  25546752,  113541120,   283852800,   408748032,   317915136,   103809024;
  8:  329472, 10543104, 110702592,  590413824,  1845043200,  3542482944,  4132896768,  2699034624,   759103488;
  9: 1025024, 36900864, 442810368, 2755264512, 10332241920, 24797380608, 38573703168, 37786484736, 21254897664, 5248122880;
%F Formula
The row polynomials satisfy (n+2*n^2*x)*P(n,x) = (2*n+10+(4*n^2+28*n+40)*x+(8*n^2+48*n+40)*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[((2*n + 10 + (4*n^2 + 28*n + 40)*x + (8*n^2 + 48*n + 40)*x^2)*P[n - 1, x])/(n + 2*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] = ((2*n + 10 + (4*n^2 + 28*n + 40)*x + (8*n^2 + 48*n + 40)*x^2)*v[i - 1])/(n + 2*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.