Real-rooted atlas

A143491 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 9\,x + 5\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 12 + 31\,x + 20\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
012312063
160227286120693
23601782330527228409009
32520153543739145494276566720135135
420160145512437314700471630694302664604802297295
51814401511208539094610677379126809569030916357096060480043648605
618144001710792070532748166074196244260729229799676135049676453285606652800916620705
719958400209959920981155268267320913246797583715458925184424314254821192102726171235207983360021082276215

All 15 rows

Technical details
Node id
n-c9d123988faa32c3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4123153381f31efd

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         12,          31,           20;
  1:         60,         227,          286,          120;
  2:        360,        1782,         3305,         2722,           840;
  3:       2520,       15354,        37391,        45494,         27656,          6720;
  4:      20160,      145512,       437314,       700471,        630694,        302664,         60480;
  5:     181440,     1511208,      5390946,     10677379,      12680956,       9030916,       3570960,        604800;
  6:    1814400,    17107920,     70532748,    166074196,     244260729,     229799676,     135049676,      45328560,       6652800;
  7:   19958400,   209959920,    981155268,   2673209132,    4679758371,    5458925184,    4243142548,    2119210272,     617123520,     79833600;
  8:  239500800,  2778978240,  14503342176,  44833528068,   90908819168,  126343961031,  121883737968,   80591376388,   34955215776,   8980608960,   1037836800;
  9: 3113510400, 39479728320, 227449143648, 785882655348, 1809484042136, 2915194961755, 3353304048018, 2754060224596, 1582697074520, 606120937344, 139220403840, 14529715200;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+4+(2*n+9)*x+(n+5)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 31*x + 20*x^2.
%t Mathematica
P[0, x_] := 12 + 31*x + 20*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (2*n + 9)*x + (n + 5)*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] = 12 + 31*x + 20*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (2*n + 9)*x + (n + 5)*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

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.