Real-rooted atlas

A322944 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 + 10\,x - x^{2} + 7\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(15\,x - 3\,x^{2} - 3\,n x\right)\,P_{n-2}'(x) + \left(-6 + 2\,x + 2\,n\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
3212317
4676454131
52454456411251249
6120440070202320225614091
772039840908404416069503967182913
8504039984012406808276801872501705263782678187

All 15 rows

Technical details
Node id
n-5d0f3e585742a565 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-260aff1e54e87d47

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,        2;
   3:      2,       12,         3;
   4:      6,       76,        45,         4;
   5:     24,      544,       564,       112,         5;
   6:    120,     4400,      7020,      2320,       225,        6;
   7:    720,    39840,     90840,     44160,      6950,      396,       7;
   8:   5040,   399840,   1240680,    827680,    187250,    17052,     637,     8;
   9:  40320,  4408320,  17969280,  15724800,   4779600,   616896,   36456,   960,    9;
  10: 362880, 52980480, 276151680, 307005440, 120027600, 20541024, 1704024, 70464, 1377, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+10+(2*n-10)*x+x^2)*P(n-2,x) - ((3*n-15)*x+3*x^2)*d/dx P(n-2,x) + (2*n-6+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 10 + (2*n - 10)*x + x^2)*P[n - 2, x] - ((3*n - 15)*x + 3*x^2)*D[P[n - 2, x], x] + (2*n - 6 + 2*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
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 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 7*n + 10 + (2*n - 10)*x + x^2)*v[i - 2] - ((3*n - 15)*x + 3*x^2)*deriv(v[i - 2], x) + (2*n - 6 + 2*x)*v[i - 1] + 3*x*deriv(v[i - 1], 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.