Real-rooted atlas

A080248 → D

nonnegative coefficients

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(-6 - 4\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 5\,x^{2}\right)\,P_{n-2}'(x) + \left(-x^{2} - x^{3}\right)\,P_{n-2}''(x) + \left(8 + 4\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2426
31320336
440146604250
512195681914052041
6364597896243192280619444
71093365961052076071298705047213989
83280221762110696410509562831602595684082692926
998411337132114022831724029272600551066560605221320938378014
10295248042474116028528273807424174011520385757463423112128568198010614048886
1188573483138921172348991426181606439965012001288934712168237069969768825353928601110946194599
1226572029006049811795960748654903729488925379528040885973376756029164862866689624854544470470400412215930716144
13797161174089442811839469822799808354188419561926670651251911808864319870801600369303436322074250178586671808298295460134685259305521
14239148410446960890118655832391215129112725896423276504357203740096184867012975912634608202743209384015563039637661767563001292605601402128728014111220015237678
Technical details
Node id
n-8bcbf46a7e9fa9da · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9bae32a71a104432

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:    4,       2;
  3:   13,      20,        3;
  4:   40,     146,       60,        4;
  5:  121,     956,      819,      140,       5;
  6:  364,    5978,     9624,     3192,     280,       6;
  7: 1093,   36596,   105207,    60712,    9870,     504,     7;
  8: 3280,  221762,  1106964,  1050956,  283160,   25956,   840,    8;
  9: 9841, 1337132, 11402283, 17240292, 7260055, 1066560, 60522, 1320, 9;
%F Formula
The row polynomials satisfy 2*P(n,x) = -(6+4*x+2*x^2)*P(n-2,x) - (6*x+5*x^2)*d/dx P(n-2,x) - (x^2+x^3)*d^2/dx^2 P(n-2,x) + (8+4*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) + 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[(-(6 + 4*x + 2*x^2)*P[n - 2, x] - (6*x + 5*x^2)*D[P[n - 2, x], x] - (x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (8 + 4*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}])/(2)]];
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] = (-(6 + 4*x + 2*x^2)*v[i - 2] - (6*x + 5*x^2)*deriv(v[i - 2], x) - (x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (8 + 4*x)*v[i - 1] + 6*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x))/(2)); 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.