Real-rooted atlas

A173018 → x ↦ x+1 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 + 2\,x + n\right)\,P_{n}(x) = \left(6 + 2\,x - 21\,x^{2} - 17\,x^{3} + 6\,x^{4} - 11\,n - 15\,n x + 10\,n x^{2} + 14\,n x^{3} - 2\,n x^{4} + 6\,n^{2} + 11\,n^{2} x + 2\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 7\,x^{3} - 3\,x^{4} + 2\,x^{5} - 3\,n x - 4\,n x^{2} + 2\,n x^{3} + 3\,n x^{4} + n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(6 + 2\,x - 13\,x^{2} + 2\,x^{3} - 7\,n - 5\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{3} - n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
211
3314
4128121
56064171142
63605402403411175
725204920318078867111476
82016048720420001562424361321129073
918144052416056952028929669804732026111641802
101814400610848080438405241600176652029538021720518123292459
11199584007680960011914560095376960419504401001412012103806415610311364530688
12239500800103783680018561312001768979520968647680306876240540936004866048189420205616237123365
13311351040015008716800304365600003373102656022222347840888025248021117254402829361921934842856009641051115806988342
144358914560023143760640052514542080066452690304051341786496024942677760075785149440139195056001448046600764782201659840820212318774566303
1565383718400037922556672009524228313600135634891392001204876632960069225444288002583521740800614013079680889552263607303023000301473900493040416395149799220552940
Technical details
Node id
n-a3af9d49eedeac96 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-08aabc8d9547846a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:       1;
   3:       3,       1;
   4:      12,       8,       1;
   5:      60,      64,      17,       1;
   6:     360,     540,     240,      34,       1;
   7:    2520,    4920,    3180,     788,      67,      1;
   8:   20160,   48720,   42000,   15624,    2436,    132,     1;
   9:  181440,  524160,  569520,  289296,   69804,   7320,   261,   1;
  10: 1814400, 6108480, 8043840, 5241600, 1766520, 295380, 21720, 518, 1;
%F Formula
The row polynomials satisfy (n-2+2*x)*P(n,x) = -(n^3-6*n^2+11*n-6+(2*n^3-11*n^2+15*n-2)*x+(n^3-2*n^2-10*n+21)*x^2+(3*n^2-14*n+17)*x^3+(2*n-6)*x^4)*P(n-2,x) + ((n^2-3*n+2)*x+(2*n^2-4*n)*x^2+(n^2+2*n-7)*x^3+(3*n-3)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2-7*n+6+(2*n^2-5*n+2)*x+(5*n-13)*x^2+2*x^3)*P(n-1,x) - ((n-2)*x+n*x^2+2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 6*n^2 + 11*n - 6 + (2*n^3 - 11*n^2 + 15*n - 2)*x + (n^3 - 2*n^2 - 10*n + 21)*x^2 + (3*n^2 - 14*n + 17)*x^3 + (2*n - 6)*x^4)*P[n - 2, x] + ((n^2 - 3*n + 2)*x + (2*n^2 - 4*n)*x^2 + (n^2 + 2*n - 7)*x^3 + (3*n - 3)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - 7*n + 6 + (2*n^2 - 5*n + 2)*x + (5*n - 13)*x^2 + 2*x^3)*P[n - 1, x] - ((n - 2)*x + n*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n - 2 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n^3 - 6*n^2 + 11*n - 6 + (2*n^3 - 11*n^2 + 15*n - 2)*x + (n^3 - 2*n^2 - 10*n + 21)*x^2 + (3*n^2 - 14*n + 17)*x^3 + (2*n - 6)*x^4)*v[i - 2] + ((n^2 - 3*n + 2)*x + (2*n^2 - 4*n)*x^2 + (n^2 + 2*n - 7)*x^3 + (3*n - 3)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - 7*n + 6 + (2*n^2 - 5*n + 2)*x + (5*n - 13)*x^2 + 2*x^3)*v[i - 1] - ((n - 2)*x + n*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n - 2 + 2*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.