Real-rooted atlas

A109983 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2} x - 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(-2\,x^{3} - 8\,n x^{3} + 3\,n^{2} x + 6\,n^{2} x^{2} - 8\,n^{2} x^{3} + 2\,n^{3} x + 8\,n^{3} x^{2} - n^{4} x\right)\,P_{n-1}(x) + \left(-x^{3} - x^{4} - 4\,n x^{3} - 4\,n x^{4} + 2\,n^{2} x^{2} - 2\,n^{2} x^{3} - 4\,n^{2} x^{4} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1041822
200996150255
30001630010809802376
400002572044108960567019785
50000036147013440453606300030492153798
6000000492688340201680003811503991681561561141231
70000000644536756005082001663200281080823543527722008188960

All 15 rows

Technical details
Node id
n-7e6005fa7072f45e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-be5fa67dc32b0b24

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 0, 4, 18;
  2: 0, 0,  9, 96, 150;
  3: 0, 0,  0, 16, 300, 1080,  980;
  4: 0, 0,  0,  0,  25,  720, 4410,  8960,  5670;
  5: 0, 0,  0,  0,   0,   36, 1470, 13440, 45360,  63000,  30492;
  6: 0, 0,  0,  0,   0,    0,   49,  2688, 34020, 168000, 381150,  399168,  156156;
  7: 0, 0,  0,  0,   0,    0,    0,    64,  4536,  75600, 508200, 1663200, 2810808,  2354352,   772200;
  8: 0, 0,  0,  0,   0,    0,    0,     0,    81,   7200, 152460, 1330560, 5855850, 14126112, 18918900,  13178880,   3719430;
  9: 0, 0,  0,  0,   0,    0,    0,     0,     0,    100,  10890,  285120, 3123120, 17657640, 56756700, 107627520, 119021760, 70887960, 17551820;
%F Formula
The row polynomials satisfy (n^4-(2*n^3-n^2)*x)*P(n,x) = -((n^4-2*n^3-3*n^2)*x-(8*n^3+6*n^2)*x^2+(8*n^2+8*n+2)*x^3)*P(n-1,x) + ((2*n^3+2*n^2)*x^2+(2*n^3-2*n^2-4*n-1)*x^3-(4*n^2+4*n+1)*x^4)*d/dx 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^4 - 2*n^3 - 3*n^2)*x - (8*n^3 + 6*n^2)*x^2 + (8*n^2 + 8*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 + 2*n^2)*x^2 + (2*n^3 - 2*n^2 - 4*n - 1)*x^3 - (4*n^2 + 4*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - (2*n^3 - n^2)*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^4 - 2*n^3 - 3*n^2)*x - (8*n^3 + 6*n^2)*x^2 + (8*n^2 + 8*n + 2)*x^3)*v[i - 1] + ((2*n^3 + 2*n^2)*x^2 + (2*n^3 - 2*n^2 - 4*n - 1)*x^3 - (4*n^2 + 4*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - (2*n^3 - n^2)*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. [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.