Real-rooted atlas

A008292 → γ (scale 1/4) (shift +1) → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
28412
3403272
424027248560
5168024648164960
6134402419211904108850624
712096025804816896031744579712
8120960029836802442240707584396807382784
9133056003725568036622080144404481768960103392768
101596672005002905605742950402874685445592064021227521579764736
112075673600719566848094492569605744234496154918400013420953626148227072
1229059430400110425835520163286323200116914237440405490176005711271936156577792466102693888
134358914560001801684684800296267811840024432062054401039052851200206057828352133263011848901897445376
14697426329600031151709388800563919003648005265458085888026573318553600685249059225673452870041615230058496181348021813248
Technical details
Node id
n-76aeb61aa1ba0ef0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-44df5437d7babb90

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:        2;
  2:        8,        4;
  3:       40,       32;
  4:      240,      272,       48;
  5:     1680,     2464,      816;
  6:    13440,    24192,    11904,     1088;
  7:   120960,   258048,   168960,    31744;
  8:  1209600,  2983680,  2442240,   707584,   39680;
  9: 13305600, 37255680, 36622080, 14440448, 1768960;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-n-2+(2*n^2-2*n-4)*x+(n^2-n-2)*x^2)*P(n-2,x) + ((2*n+2)*x+(4*n+4)*x^2+(2*n+2)*x^3)*d/dx P(n-2,x) + (2*n+(2*n-2)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(n^2 - n - 2 + (2*n^2 - 2*n - 4)*x + (n^2 - n - 2)*x^2)*P[n - 2, x] + ((2*n + 2)*x + (4*n + 4)*x^2 + (2*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n + (2*n - 2)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - n - 2 + (2*n^2 - 2*n - 4)*x + (n^2 - n - 2)*x^2)*v[i - 2] + ((2*n + 2)*x + (4*n + 4)*x^2 + (2*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n + (2*n - 2)*x)*v[i - 1] - (2*x + 2*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 - 1, v[i])))};
T(9)

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.