Real-rooted atlas

A008292 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
3138122
47567171160
55416342463411456
646836793356279867115904
74729381780541111690424511321202672
8545835109596787625535237173649734126112951680

All 15 rows

Technical details
Node id
n-76e39d06b3685c1c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6c43fbf0495b302d

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:         3,         1;
   3:        13,         8,         1;
   4:        75,        67,        17,         1;
   5:       541,       634,       246,        34,        1;
   6:      4683,      6793,      3562,       798,       67,        1;
   7:     47293,     81780,     54111,     16904,     2451,      132,       1;
   8:    545835,   1095967,    876255,    352371,    73649,     7341,     261,     1;
   9:   7087261,  16205198,  15196348,   7487306,  2056690,   306146,   21748,   518,    1;
  10: 102247563, 262283125, 282270272, 164793976, 56259206, 11247058, 1239080, 64192, 1031, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(n-1)*x)*P(n-1,x) - (2+3*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + (n - 1)*x)*P[n - 1, x] - (2 + 3*x + x^2)*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; for(n = 2, N, my(i = n); v[i] = (2*n - 1 + (n - 1)*x)*v[i - 1] - (2 + 3*x + x^2)*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

Same fingerprint ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.