Real-rooted atlas

A062145 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2459
310302161
42010516884377
5352807568403302241
656630252046203960128713073
78412606930184802574018018500575517
8120231016632600601201201351358008019448433905
9165396036036168168450450720720680680350064755822485825
102206435720724204201441440306306040840803325608151164029393014218905
11286100101351359609604084080110270161939938022170720158722206466460114406681270333
1236415015240240204204010501920349188847759752011639628011639628074364290274575844457400464387817
134552184040840840840802494206099768240271591320512143632669278610594914320343219800115892400173838602653649025
1456030940668304775975255426800261891630853572720196321725632125373283718214500297457160015645474004867480806786391515167050785
156804284010581481410864011639628064017954024540215706731030592133855722001933471540020078358300146024424007057847160203591745026518252586716873725
Technical details
Node id
n-0b7abf4bfe5d62d7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b9e11a0a9cd0a58e

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 -1)
  • 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:   4,    5;
   3:  10,   30,    21;
   4:  20,  105,   168,     84;
   5:  35,  280,   756,    840,     330;
   6:  56,  630,  2520,   4620,    3960,    1287;
   7:  84, 1260,  6930,  18480,   25740,   18018,    5005;
   8: 120, 2310, 16632,  60060,  120120,  135135,   80080,   19448;
   9: 165, 3960, 36036, 168168,  450450,  720720,  680680,  350064,   75582;
  10: 220, 6435, 72072, 420420, 1441440, 3063060, 4084080, 3325608, 1511640, 293930;
%F Formula
The row polynomials satisfy (n^2+n-2)*P(n,x) = (n^2+4*n+4+(2*n^2+5*n+2)*x)*P(n-1,x) + ((2*n+1)*x+(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 4 + (2*n^2 + 5*n + 2)*x)*P[n - 1, x] + ((2*n + 1)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 + 4*n + 4 + (2*n^2 + 5*n + 2)*x)*v[i - 1] + ((2*n + 1)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + n - 2)); 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

Candidate equalities ?

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.