Real-rooted atlas

A286718 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 + x + 2\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
210414091
3804387714401729
488059381461315634616043225
512320980923055284672973520181047201339975
62094401913964715581614054609153302468820600209440049579075
7418880043096400187137492445675948629860927529007658244761800481712002131900225

All 15 rows

Technical details
Node id
n-68a56457f6a51842 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f17bd99bf272942d

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:          2,           5;
  2:         10,          41,           40;
  3:         80,         438,          771,          440;
  4:        880,        5938,        14613,        15634,          6160;
  5:      12320,       98092,       305528,       467297,        352018,        104720;
  6:     209440,     1913964,      7155816,     14054609,      15330246,       8820600,       2094400;
  7:    4188800,    43096400,    187137492,    445675948,     629860927,     529007658,     244761800,     48171200;
  8:   96342400,  1100126000,   5424668716,  15116121596,   26074375969,   28543560236,   19383720508,   7471744400,   1252451200;
  9: 2504902400, 31397205600, 172945040616, 550334554260, 1116301301478, 1498289469237, 1331739980052, 756393249132, 249244318800, 36321084800;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3*n-1+(6*n+1)*x+(3*n+2)*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[((3*n - 1 + (6*n + 1)*x + (3*n + 2)*x^2)*P[n - 1, x])/(1 + 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] = ((3*n - 1 + (6*n + 1)*x + (3*n + 2)*x^2)*v[i - 1])/(1 + 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

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.