Real-rooted atlas

A371259 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(26 + x + 10\,n + n^{2}\right)\,P_{n-1}(x) + \left(11 + 11\,x + 2\,n + 2\,n x\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
137138
2186510211968
3122961102672031133432
410278281116846035102348111482192
5106271424515444718162843629437054511223540704
6133215991777237723398101230439231249384922176958021158243927808
71991308818137742411071089032683411294616893548643810184674504851127124429511438464

All 15 rows

Technical details
Node id
n-f6bd524925313b17 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a8bb4bb33283a6b7

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:                 37,                  1;
  2:               1865,                102,                 1;
  3:             122961,              10267,               203,               1;
  4:           10278281,            1168460,             35102,             348,              1;
  5:         1062714245,          154447181,           6284362,           94370,            545,            1;
  6:       133215991777,        23772339810,        1230439231,        24938492,         217695,          802,         1;
  7:     19913088181377,      4241107108903,      268341129461,      6893548643,       81018467,       450485,      1127,       1;
  8:   3500271565033393,    871016335435864,    65528274612492,   2050686578056,    30467772758,    227676328,    858732,    1528,    1;
  9: 714944028588443461, 204425020518981497, 17915166495346948, 664497423961412, 11994858607046, 113087610958, 572466452, 1534452, 2013, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^2+10*n+26+x)*P(n-1,x) + (2*n+11+(2*n+11)*x)*d/dx P(n-1,x) + (1+2*x+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[(n^2 + 10*n + 26 + x)*P[n - 1, x] + (2*n + 11 + (2*n + 11)*x)*D[P[n - 1, x], x] + (1 + 2*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; for(n = 1, N, my(i = n + 1); v[i] = (n^2 + 10*n + 26 + x)*v[i - 1] + (2*n + 11 + (2*n + 11)*x)*deriv(v[i - 1], x) + (1 + 2*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.