Real-rooted atlas

A106800 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(10 + 32\,x + 24\,x^{2} - 15\,n - 40\,n x - 26\,n x^{2} + 7\,n^{2} + 16\,n^{2} x + 9\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4 + 14\,x + 16\,x^{2} + 6\,x^{3} - 4\,n - 13\,n x - 14\,n x^{2} - 5\,n x^{3} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(13 + 18\,x - 10\,n - 12\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(3 + 6\,x + 3\,x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
10
20
311
4527
52320346
6109152574322
7544106267214452427
82876728266122524345619645
91611350416602673388487458047170236
109549535712853289340068415665529028184181573732
115971552604902467936444378922359135682842941784160915459637
12392924319624376413419684755120032229115129069002874942302024929710160769075
1327132324152895932370327284502269332415996730214799976674261981186012096345920580111763691946
14196122796123231994833789219725291287148519668363033046153201353152794342004528483680253067360451771220346588710
1514805312851027230264031490793474560132813286379063833548378476724245936600848207808760170269713019611307097607409844813246136162031
Technical details
Node id
n-09fbd62b1b37c2d7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-53105ee0eb8f88ac

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
  • 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:     0;
   2:     0;
   3:     1;
   4:     5,      2;
   5:    23,     20,      3;
   6:   109,    152,     57,      4;
   7:   544,   1062,    672,    144,      5;
   8:  2876,   7282,   6612,   2524,    345,     6;
   9: 16113,  50416,  60267,  33884,   8745,   804,    7;
  10: 95495, 357128, 532893, 400684, 156655, 29028, 1841, 8;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(n^3-7*n^2+15*n-10+(2*n^3-16*n^2+40*n-32)*x+(n^3-9*n^2+26*n-24)*x^2)*P(n-2,x) + (n^2-4*n+4+(3*n^2-13*n+14)*x+(3*n^2-14*n+16)*x^2+(n^2-5*n+6)*x^3)*d/dx P(n-2,x) + (2*n^2-10*n+13+(2*n^2-12*n+18)*x)*P(n-1,x) - (n-3+(2*n-6)*x+(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 7*n^2 + 15*n - 10 + (2*n^3 - 16*n^2 + 40*n - 32)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*P[n - 2, x] + (n^2 - 4*n + 4 + (3*n^2 - 13*n + 14)*x + (3*n^2 - 14*n + 16)*x^2 + (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 10*n + 13 + (2*n^2 - 12*n + 18)*x)*P[n - 1, x] - (n - 3 + (2*n - 6)*x + (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = (-(n^3 - 7*n^2 + 15*n - 10 + (2*n^3 - 16*n^2 + 40*n - 32)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*v[i - 2] + (n^2 - 4*n + 4 + (3*n^2 - 13*n + 14)*x + (3*n^2 - 14*n + 16)*x^2 + (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 10*n + 13 + (2*n^2 - 12*n + 18)*x)*v[i - 1] - (n - 3 + (2*n - 6)*x + (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.