Real-rooted atlas

A166961 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21111123
3103146451295
414422677139315815671
5280236382346265109755121149599
67128051917362178164065094075450159315139791
722857916707811358001560239502740787687547137548591221510503

All 15 rows

Technical details
Node id
n-59fb022e36d2469d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ffa7f44dd08a8e3c

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:           2,            1;
  2:          11,           11,            1;
  3:         103,          146,           45,            1;
  4:        1442,         2677,         1393,          158,           1;
  5:       28023,        63823,        46265,        10975,         512,           1;
  6:      712805,      1917362,      1781640,       650940,       75450,        1593,         1;
  7:    22857916,     70781135,     80015602,     39502740,     7876875,      471375,      4859,        1;
  8:   898132469,   3140399343,   4173133419,   2596224062,   747179325,    84562626,   2755502,    14684,     1;
  9: 42280116147, 164514396100, 250868890047, 188139393033, 71127493222, 12433552095, 826175938, 15378314, 44190, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+(2*n-1)*x)*P(n-1,x) + (3+x-2*x^2)*d/dx P(n-1,x) + (2+4*x+2*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + (2*n - 1)*x)*P[n - 1, x] + (3 + x - 2*x^2)*D[P[n - 1, x], x] + (2 + 4*x + 2*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; for(n = 1, N, my(i = n + 1); v[i] = (2*n + (2*n - 1)*x)*v[i - 1] + (3 + x - 2*x^2)*deriv(v[i - 1], x) + (2 + 4*x + 2*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.