Real-rooted atlas

A390433 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-24 - 34\,x - 13\,x^{2} - x^{3} + 44\,n + 41\,n x + 8\,n x^{2} - 30\,n^{2} - 16\,n^{2} x - n^{2} x^{2} + 9\,n^{3} + 2\,n^{3} x - n^{4}\right)\,P_{n-2}(x) + \left(18\,x + 19\,x^{2} + 5\,x^{3} - 27\,n x - 18\,n x^{2} - 2\,n x^{3} + 13\,n^{2} x + 4\,n^{2} x^{2} - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-6\,x^{2} - 5\,x^{3} - x^{4} + 5\,n x^{2} + 2\,n x^{3} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(7 + 4\,x + x^{2} - 7\,n - 2\,n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
346111
43676231136
557616487046012989
614400541442971237601251102142
7518400249465616775042817921410022614986679
825401600152918784122562432257204481727892418043711328373332

All 15 rows

Technical details
Node id
n-d8fa2923d2fa07ba · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-29c4e8a40999cf06

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:            1;
   2:            1,             1;
   3:            4,             6,             1;
   4:           36,            76,            23,            1;
   5:          576,          1648,           704,           60,           1;
   6:        14400,         54144,         29712,         3760,         125,          1;
   7:       518400,       2494656,       1677504,       281792,       14100,        226,        1;
   8:     25401600,     152918784,     122562432,     25720448,     1727892,      41804,      371,      1;
   9:   1625702400,   12005397504,   11261507328,   2842711040,   240881472,    7841888,   105056,    568,   1;
  10: 131681894400, 1172902809600, 1270850833152, 376086392064, 38638733120, 1589779648, 28647680, 233664, 825, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-9*n^3+30*n^2-44*n+24-(2*n^3-16*n^2+41*n-34)*x+(n^2-8*n+13)*x^2+x^3)*P(n-2,x) - ((2*n^3-13*n^2+27*n-18)*x-(4*n^2-18*n+19)*x^2+(2*n-5)*x^3)*d/dx P(n-2,x) - ((n^2-5*n+6)*x^2-(2*n-5)*x^3+x^4)*d^2/dx^2 P(n-2,x) + (2*n^2-7*n+7-(2*n-4)*x+x^2)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 9*n^3 + 30*n^2 - 44*n + 24 - (2*n^3 - 16*n^2 + 41*n - 34)*x + (n^2 - 8*n + 13)*x^2 + x^3)*P[n - 2, x] - ((2*n^3 - 13*n^2 + 27*n - 18)*x - (4*n^2 - 18*n + 19)*x^2 + (2*n - 5)*x^3)*D[P[n - 2, x], x] - ((n^2 - 5*n + 6)*x^2 - (2*n - 5)*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 7*n + 7 - (2*n - 4)*x + x^2)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x] + x^2*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] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = -(n^4 - 9*n^3 + 30*n^2 - 44*n + 24 - (2*n^3 - 16*n^2 + 41*n - 34)*x + (n^2 - 8*n + 13)*x^2 + x^3)*v[i - 2] - ((2*n^3 - 13*n^2 + 27*n - 18)*x - (4*n^2 - 18*n + 19)*x^2 + (2*n - 5)*x^3)*deriv(v[i - 2], x) - ((n^2 - 5*n + 6)*x^2 - (2*n - 5)*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 7*n + 7 - (2*n - 4)*x + x^2)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x) + x^2*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.