Real-rooted atlas

A166972 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-8\,n + 4\,n x - 2\,n x^{2} - 2\,n^{2} x + 3\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-13\,n x + 11\,n x^{2} - n x^{3} - 10\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-3\,n x^{2} + 3\,n x^{3} - 3\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-8 + 2\,x + 9\,n - 4\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-13\,x + x^{2} + 13\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(-3\,x^{2} + 3\,n x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
210212
383823168
4668222036643258
5534951964246301236583184
64279811447341296984178860357062666952
7342391245529906046486518543148101344593667104926212

All 15 rows

Technical details
Node id
n-3d472a94c673517d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b4f90a15b05d202b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        1;
  2:       10,           2;
  3:       83,          82,            3;
  4:      668,        2220,          366,            4;
  5:     5349,       51964,        24630,         1236,           5;
  6:    42798,     1144734,      1296984,       178860,        3570,          6;
  7:   342391,    24552990,     60464865,     18543148,     1013445,       9366,        7;
  8:  2739136,   520406264,   2639571510,   1608403720,   190006460,    4894080,    23086,     8;
  9: 21913097, 10972357720, 111047126172, 125662813880, 28293143410, 1562943648, 21192416, 54568, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(8*n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (13*n*x+(10*n^2-11*n)*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) - (3*n*x^2+(3*n^2-3*n)*x^3)*d^2/dx^2 P(n-2,x) + (9*n-8+(2*n^2-4*n+2)*x)*P(n-1,x) + ((13*n-13)*x-(n-1)*x^2)*d/dx P(n-1,x) + (3*n-3)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (13*n*x + (10*n^2 - 11*n)*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] - (3*n*x^2 + (3*n^2 - 3*n)*x^3)*D[P[n - 2, x], {x, 2}] + (9*n - 8 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((13*n - 13)*x - (n - 1)*x^2)*D[P[n - 1, x], x] + (3*n - 3)*x^2*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (13*n*x + (10*n^2 - 11*n)*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) - (3*n*x^2 + (3*n^2 - 3*n)*x^3)*deriv(deriv(v[i - 2], x), x) + (9*n - 8 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((13*n - 13)*x - (n - 1)*x^2)*deriv(v[i - 1], x) + (3*n - 3)*x^2*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.