Real-rooted atlas

A119308 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-2\,x^{2} + n x^{2}\right)\,P_{n-3}(x) + \left(2 + x - 4\,n - 2\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(7 + 3\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
\(P_{2}(x) = 9 + 7\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
297117
3283512176
49015688181353
52976605251802511688
610012717280513753253318257
7343211011140149009308053942141128
811934442006697653508242066188840521207905
941990176358310284297024168168573301146612486311063932
101492267009101405050156978010710004598161234801995017857515501073
1153488827784466253280799425063954003325608113097624684033000247588128695252
1219315401099492027457584395519963634719022255992920937625581604645085236033441021150827073
13702040543459650119272595191222460198658889140229804685906652327925654041138314028022344201171798054000
1425662825171655785513590805907284105105203598584206622547636889319178487954603549107837731426425119301573313314247388417
Technical details
Node id
n-03c2a02884bf8065 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7e6616e08c6108b9

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,      1;
  2:     9,      7,      1;
  3:    28,     35,     12,      1;
  4:    90,    156,     88,     18,      1;
  5:   297,    660,    525,    180,     25,     1;
  6:  1001,   2717,   2805,   1375,    325,    33,     1;
  7:  3432,  11011,  14014,   9009,   3080,   539,    42,    1;
  8: 11934,  44200,  66976,  53508,  24206,  6188,   840,   52,  1;
  9: 41990, 176358, 310284, 297024, 168168, 57330, 11466, 1248, 63, 1;
%F Formula
The row polynomials satisfy (n+3)*P(n,x) = (n-2)*x^2*P(n-3,x) - (4*n-2+(2*n-1)*x+n*x^2)*P(n-2,x) + (5*n+7+(2*n+3)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 9 + 7*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 9 + 7*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*x^2*P[n - 3, x] - (4*n - 2 + (2*n - 1)*x + n*x^2)*P[n - 2, x] + (5*n + 7 + (2*n + 3)*x)*P[n - 1, x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 9 + 7*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2)*x^2*v[i - 3] - (4*n - 2 + (2*n - 1)*x + n*x^2)*v[i - 2] + (5*n + 7 + (2*n + 3)*x)*v[i - 1])/(n + 3)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.