Real-rooted atlas

A253283 → reverse

nonnegative coefficients

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

Recurrence

\(\left(5 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-9\,x^{2} + 9\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(30 + 16\,x - 32\,n - 16\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
211
3325
41012325
53560304129
6126280210605681
74621260126056010563653
817165544693042001260168719825
964352402436036277201155025202528108545
1024310102960180180168168900902772046203609598417
1192378437580875160960960630630252252600607920495103317445
1235271618475604157010525096040840802018016630630120120128706601118474633
131352078775975219399380277134002494206014702688571771214414402252252002085812103274625
1452003003244987289237148142262120145495350997682404655851214702688306306040040030030109213579168825
1520058300135207800405623400713897184818007190640179540349188840133024320349188846126120680680436801365143256957317
Technical details
Node id
n-133f3a16dadf73ac · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-2c1fea4a558d4e6c

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     1;
   3:     3,      2;
   4:    10,     12,      3;
   5:    35,     60,     30,      4;
   6:   126,    280,    210,     60,     5;
   7:   462,   1260,   1260,    560,   105,     6;
   8:  1716,   5544,   6930,   4200,  1260,   168,    7;
   9:  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;
  10: 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;
%F Formula
The row polynomials satisfy (2*n^2-7*n+5)*P(n,x) = -(2*n^2-9*n+9)*x^2*P(n-2,x) + (8*n^2-32*n+30+(4*n^2-16*n+16)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 9*n + 9)*x^2*P[n - 2, x] + (8*n^2 - 32*n + 30 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x])/(2*n^2 - 7*n + 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 9*n + 9)*x^2*v[i - 2] + (8*n^2 - 32*n + 30 + (4*n^2 - 16*n + 16)*x)*v[i - 1])/(2*n^2 - 7*n + 5)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.