Real-rooted atlas

A049374 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36 + 12\,x + 4\,x^{2} - 13\,n + 3\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-20\,x + 20\,x^{2} - 5\,n x\right)\,P_{n-2}'(x) + \left(14 - 3\,x + 2\,n\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1718
26020181
36123483911000
472246120114064114549
596264112824307802820951242784
614192642201472819504108360588013214554613
7227969284550515222098384393758430492010920175194654064

All 15 rows

Technical details
Node id
n-4019cc30eb72d880 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2f94d007fcf36594

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 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:          7,           1;
  2:         60,          20,           1;
  3:        612,         348,          39,          1;
  4:       7224,        6120,        1140,         64,         1;
  5:      96264,      112824,       30780,       2820,        95,        1;
  6:    1419264,     2201472,      819504,     108360,      5880,      132,       1;
  7:   22796928,    45505152,    22098384,    3937584,    304920,    10920,     175,     1;
  8:  394377984,   994194432,   610860096,  140374080,  14503104,   734832,   18648,   224,   1;
  9: 7285614336, 22885099776, 17405720640, 5003570880, 660623040, 44307648, 1580040, 29880, 279, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+13*n+36-(3*n+12)*x-4*x^2)*P(n-2,x) - ((5*n+20)*x-20*x^2)*d/dx P(n-2,x) + (2*n+14-3*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 13*n + 36 - (3*n + 12)*x - 4*x^2)*P[n - 2, x] - ((5*n + 20)*x - 20*x^2)*D[P[n - 2, x], x] + (2*n + 14 - 3*x)*P[n - 1, x] + 5*x*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] = 7 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 13*n + 36 - (3*n + 12)*x - 4*x^2)*v[i - 2] - ((5*n + 20)*x - 20*x^2)*deriv(v[i - 2], x) + (2*n + 14 - 3*x)*v[i - 1] + 5*x*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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.