Real-rooted atlas

A008292 → γ (scale 1/2) (shift +1) → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3} + n + 4\,n x + 5\,n x^{2} + 2\,n x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 2\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-1 - 3\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2325
39817
435461293
5155256102513
679116029481363477
74529108008256198425569
8288397929475048270722480212733
9201939628504707478336192552801929393
10154273953624667005672415920810396006633619176021
1112767689490182327288898451886976173455602097024206004465
121137946034782388147976411406628019202791173605195889624465282385999261
1310866574034961698704917338215087173065604432102560111724857610411172829592507681
14110686048475456161651411075640183611850409524070690189520226161992643285699648118984832391601791701
Technical details
Node id
n-cf6c2be86a6239dc · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b3fb340d296c4b40

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:      1;
  2:      3,      2;
  3:      9,      8;
  4:     35,     46,     12;
  5:    155,    256,    102;
  6:    791,   1602,    948,    136;
  7:   4529,  10800,   8256,   1984;
  8:  28839,  79294,  75048,  27072,  2480;
  9: 201939, 628504, 707478, 336192, 55280;
%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) + (n+1+(4*n+4)*x+(5*n+5)*x^2+(2*n+2)*x^3)*d/dx P(n-2,x) + (2*n-1+(2*n-2)*x)*P(n-1,x) - (1+3*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
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] + (n + 1 + (4*n + 4)*x + (5*n + 5)*x^2 + (2*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n - 1 + (2*n - 2)*x)*P[n - 1, x] - (1 + 3*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] = 1; 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] + (n + 1 + (4*n + 4)*x + (5*n + 5)*x^2 + (2*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n - 1 + (2*n - 2)*x)*v[i - 1] - (1 + 3*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

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.