Real-rooted atlas

A088729 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-14 - 6\,x - x^{2}\right)\,P_{n-2}(x) + \left(-18 - 13\,x - 3\,x^{2}\right)\,P_{n-2}'(x) + \left(-6\,x - 2\,x^{2}\right)\,P_{n-2}''(x) + \left(9 + 2\,x\right)\,P_{n-1}(x) + \left(6 + 3\,x\right)\,P_{n-1}'(x) + \left(2\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 9 + 2\,x\)
\(P_{2}(x) = 79 + 36\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
19211
279363118
37655309041389
4831177401995180518231
51009891182024095056003156266062
6136243919209968367031537201309050474287459
7202464453340109017533530404972446305026964756875721567
832897247162163918038199571510635030015284535119448050610108091455488380

All 15 rows

Technical details
Node id
n-e193497bf1792138 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5063296cf612be30

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:            9,            2;
   2:           79,           36,            3;
   3:          765,          530,           90,           4;
   4:         8311,         7740,         1995,         180,           5;
   5:       100989,       118202,        40950,        5600,         315,          6;
   6:      1362439,      1920996,       836703,      153720,       13090,        504,         7;
   7:     20246445,     33401090,     17533530,     4049724,      463050,      26964,       756,       8;
   8:    328972471,    621639180,    381995715,   106350300,    15284535,    1194480,     50610,    1080,      9;
   9:   5805917469,  12364629002,   8706916350,  2838239800,   491673105,   48284082,   2744280,   88440,   1485,   10;
  10: 110645911399, 262204157556, 208177133703, 77783127840, 15777801380, 1867424328, 133337589, 5761800, 146025, 1980, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(14+6*x+x^2)*P(n-2,x) - (18+13*x+3*x^2)*d/dx P(n-2,x) - (6*x+2*x^2)*d^2/dx^2 P(n-2,x) + (9+2*x)*P(n-1,x) + (6+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 9 + 2*x, P(2,x) = 79 + 36*x + 3*x^2.
%t Mathematica
P[1, x_] := 9 + 2*x;
P[2, x_] := 79 + 36*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(14 + 6*x + x^2)*P[n - 2, x] - (18 + 13*x + 3*x^2)*D[P[n - 2, x], x] - (6*x + 2*x^2)*D[P[n - 2, x], {x, 2}] + (9 + 2*x)*P[n - 1, x] + (6 + 3*x)*D[P[n - 1, x], x] + 2*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 9 + 2*x; v[2] = 79 + 36*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = -(14 + 6*x + x^2)*v[i - 2] - (18 + 13*x + 3*x^2)*deriv(v[i - 2], x) - (6*x + 2*x^2)*deriv(deriv(v[i - 2], x), x) + (9 + 2*x)*v[i - 1] + (6 + 3*x)*deriv(v[i - 1], x) + 2*x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.