Real-rooted atlas

A008292 → θ+1 (shift +2) → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\,n - 4\,n x - 2\,n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 4\,x^{2} + 2\,x^{3} + n x + 2\,n x^{2} + n x^{3}\right)\,P_{n-2}'(x) + \left(3 + 2\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 12 + 14\,x + 3\,x^{2}\)
\(P_{2}(x) = 60 + 100\,x + 45\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11214329
260100454209
336078054012451809
42520672063002408315618269
52016063840756004082496607627210853
618144066528095256066729623310036300178582737769
7181440075600001270080010956960510300012277801306204088939497657
81995840093139200179625600184282560107692200355291206121500456016920710626813813

All 15 rows

Technical details
Node id
n-79b3d41ceab9652a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ef5562416e43169f

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 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:         12,          14,           3;
   2:         60,         100,          45,           4;
   3:        360,         780,         540,         124,           5;
   4:       2520,        6720,        6300,        2408,         315,           6;
   5:      20160,       63840,       75600,       40824,        9660,         762,          7;
   6:     181440,      665280,      952560,      667296,      233100,       36300,       1785,          8;
   7:    1814400,     7560000,    12700800,    10956960,     5103000,     1227780,     130620,       4088,         9;
   8:   19958400,    93139200,   179625600,   184282560,   107692200,    35529120,    6121500,     456016,      9207,      10;
   9:  239500800,  1237420800,  2694384000,  3205984320,  2258625600,   953029440,  231739200,   29352048,   1557468,   20470,    11;
  10: 3113510400, 17643225600, 42810768000, 57980482560, 47870222400, 24690631680, 7829902080, 1441632192, 136756620, 5232500, 45045, 12;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n-2,x) + ((n+2)*x+(2*n+4)*x^2+(n+2)*x^3)*d/dx P(n-2,x) + (2*n+3+(2*n+2)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 14*x + 3*x^2, P(2,x) = 60 + 100*x + 45*x^2 + 4*x^3.
%t Mathematica
P[1, x_] := 12 + 14*x + 3*x^2;
P[2, x_] := 60 + 100*x + 45*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^2)*P[n - 2, x] + ((n + 2)*x + (2*n + 4)*x^2 + (n + 2)*x^3)*D[P[n - 2, x], x] + (2*n + 3 + (2*n + 2)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 12 + 14*x + 3*x^2; v[2] = 60 + 100*x + 45*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = -(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^2)*v[i - 2] + ((n + 2)*x + (2*n + 4)*x^2 + (n + 2)*x^3)*deriv(v[i - 2], x) + (2*n + 3 + (2*n + 2)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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. [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.