Real-rooted atlas

A114655 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
122
28210
34020262
4224168362430
5134413444485623194
6844810560480096080224850
75491282368475201320018001082199910
836608064064044844816016030800308014021649350

All 15 rows

Technical details
Node id
n-8a7f4c5c028e757f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-77e051dbf841d62e

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:        2;
   2:        8,        2;
   3:       40,       20,        2;
   4:      224,      168,       36,        2;
   5:     1344,     1344,      448,       56,       2;
   6:     8448,    10560,     4800,      960,      80,       2;
   7:    54912,    82368,    47520,    13200,    1800,     108,      2;
   8:   366080,   640640,   448448,   160160,   30800,    3080,    140,    2;
   9:  2489344,  4978688,  4100096,  1793792,  448448,   64064,   4928,  176,   2;
  10: 17199104, 38697984, 36661248, 19009536, 5870592, 1100736, 122304, 7488, 216, 2;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (8*n-4+(3*n-3)*x)*P(n-1,x) - (4*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 4 + (3*n - 3)*x)*P[n - 1, x] - (4*x + 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((8*n - 4 + (3*n - 3)*x)*v[i - 1] - (4*x + 2*x^2)*deriv(v[i - 1], x))/(n + 1)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.