Real-rooted atlas

A371081 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 2\,x - x^{2} - 12\,n - 4\,n x - 13\,n^{2} - 2\,n^{2} x - 6\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-5\,x - 4\,x^{2} - 12\,n x - 2\,n x^{2} - 9\,n^{2} x - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-x^{2} - x^{3} - 2\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(10 + 2\,x + 8\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(7\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 16 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
116218
24001043507
3144005896348420648
47056004061443703286451149645
5451584003430886443124161539201800683935406
63657830400352120320057326563227914368496600333677780713543
7365783040000433381248000878320558085481882368132226560134652056848892611804948

All 15 rows

Technical details
Node id
n-17d8a24899450a15 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-870719197d2d6bee

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:               16,                 2;
  2:              400,               104,                3;
  3:            14400,              5896,              348,               4;
  4:           705600,            406144,            37032,             864,              5;
  5:         45158400,          34308864,          4312416,          153920,           1800,            6;
  6:       3657830400,        3521203200,        573265632,        27914368,         496600,         3336,          7;
  7:     365783040000,      433381248000,      87832055808,      5481882368,      132226560,      1346520,       5684,       8;
  8:   44259747840000,    63138465792000,   15493689303552,   1191558351872,    36603328960,    503380992,    3213616,    9088,     9;
  9: 6373403688960000, 10758920214528000, 3131470802184192, 288758881575936, 10859900153600, 189401101440, 1628489408, 6953472, 13824, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4+6*n^3+13*n^2+12*n+4+(2*n^2+4*n+2)*x+x^2)*P(n-2,x) - ((2*n^3+9*n^2+12*n+5)*x+(2*n+4)*x^2)*d/dx P(n-2,x) - ((n^2+2*n+1)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+8*n+10+2*x)*P(n-1,x) + (2*n+7)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 16 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 16 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (2*n^2 + 4*n + 2)*x + x^2)*P[n - 2, x] - ((2*n^3 + 9*n^2 + 12*n + 5)*x + (2*n + 4)*x^2)*D[P[n - 2, x], x] - ((n^2 + 2*n + 1)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 8*n + 10 + 2*x)*P[n - 1, x] + (2*n + 7)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
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] = 16 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (2*n^2 + 4*n + 2)*x + x^2)*v[i - 2] - ((2*n^3 + 9*n^2 + 12*n + 5)*x + (2*n + 4)*x^2)*deriv(v[i - 2], x) - ((n^2 + 2*n + 1)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 8*n + 10 + 2*x)*v[i - 1] + (2*n + 7)*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 - 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.