Real-rooted atlas

A008955 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
2105985154
3100960303331457238
4170022820113961250610204425593516
544200765020526780618025921306266602064692575376532
6163540034714740305836722143079094737469449655204801925299380412513718528824
78177000020579108002213063988013178938158446921306480999838825435511750361854755897794126253388476619528

All 15 rows

Technical details
Node id
n-8378a9feb3237cd5 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f6632b3a6a00b47e

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,             59,              85;
  3:          100,            960,            3033,             3145;
  4:         1700,          22820,          113961,           250610,           204425;
  5:        44200,         765020,         5267806,         18025921,         30626660,          20646925;
  6:      1635400,       34714740,       305836722,       1430790947,       3746944965,        5204801925,        2993804125;
  7:     81770000,     2057910800,     22130639880,     131789381584,     469213064809,      998388254355,     1175036185475,       589779412625;
  8:   5315050000,   154779092000,   1967374667800,   14253884252120,   64368720279673,   185482994188988,   332963133425110,    340319961487700,    151573309044625;
  9: 435834100000, 14419276794000, 211627927659600, 1808215275708840, 9910747444872186, 36129439614390741, 87584950052280120, 136119255205152150, 123032998825161750, 49261325439503125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n^2+1+(5*n^2+2)*x+(4*n^2+1)*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[((n^2 + 1 + (5*n^2 + 2)*x + (4*n^2 + 1)*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] = ((n^2 + 1 + (5*n^2 + 2)*x + (4*n^2 + 1)*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.