Real-rooted atlas

A173018 → reverse (shift +1) → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(n x + 3\,n x^{2} + 3\,n x^{3} + n x^{4} + n^{2} x + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-2}(x) + \left(x + 5\,x^{2} + 10\,x^{3} + 10\,x^{4} + 5\,x^{5} + x^{6} + n x + 4\,n x^{2} + 6\,n x^{3} + 4\,n x^{4} + n x^{5} - n^{2} x^{2} - 4\,n^{2} x^{3} - 6\,n^{2} x^{4} - 4\,n^{2} x^{5} - n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-1 - 4\,x - 6\,x^{2} - 4\,x^{3} - x^{4} - n - 3\,n x - 3\,n x^{2} - n x^{3} + n^{2} x + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(n x + 4\,n x^{2} + 6\,n x^{3} + 4\,n x^{4} + n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1022
20257
302161836
4023911684241
502865109004801978
602181191261807680324019195
702372659434776756007224025200214784
802755216601751405927049626407459202217602720581
90215226903082350040723209999360129124808406720217728038462214
100230572157443700740257068808953308017133984018325440010281600023587200600156943
1102612866561016121952153159600726968880193380264030210364802754259200135717120027941760010242609292
120212271203568468707716875759808551251008019578430080418973385605518364544043808688000192398976003592512000189759537241
13022455861889102882061004859094240397712515201834664832005164967808009215990784001047903816960736345209600291632140800498161664003792184441490
140249133187386961194937380263600064002764382821201623563061120584832568320013575251289600207302711616002071937387520013054949107200470762772480074101547520081304389391651
Technical details
Node id
n-8a2621cd123a76b4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c44fa03801d64ba7

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: 0, 2;
  2: 0, 2,    5;
  3: 0, 2,   16,    18;
  4: 0, 2,   39,   116,     84;
  5: 0, 2,   86,   510,    900,     480;
  6: 0, 2,  181,  1912,   6180,    7680,    3240;
  7: 0, 2,  372,  6594,  34776,   75600,   72240,    25200;
  8: 0, 2,  755, 21660, 175140,  592704,  962640,   745920,  221760;
  9: 0, 2, 1522, 69030, 823500, 4072320, 9999360, 12912480, 8406720, 2177280;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = ((n^2+n)*x+(3*n^2+3*n)*x^2+(3*n^2+3*n)*x^3+(n^2+n)*x^4)*P(n-2,x) + ((n+1)*x-(n^2-4*n-5)*x^2-(4*n^2-6*n-10)*x^3-(6*n^2-4*n-10)*x^4-(4*n^2-n-5)*x^5-(n^2-1)*x^6)*d/dx P(n-2,x) - (n+1-(n^2-3*n-4)*x-(3*n^2-3*n-6)*x^2-(3*n^2-n-4)*x^3-(n^2-1)*x^4)*P(n-1,x) + (n*x+4*n*x^2+6*n*x^3+4*n*x^4+n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(((n^2 + n)*x + (3*n^2 + 3*n)*x^2 + (3*n^2 + 3*n)*x^3 + (n^2 + n)*x^4)*P[n - 2, x] + ((n + 1)*x - (n^2 - 4*n - 5)*x^2 - (4*n^2 - 6*n - 10)*x^3 - (6*n^2 - 4*n - 10)*x^4 - (4*n^2 - n - 5)*x^5 - (n^2 - 1)*x^6)*D[P[n - 2, x], x] - (n + 1 - (n^2 - 3*n - 4)*x - (3*n^2 - 3*n - 6)*x^2 - (3*n^2 - n - 4)*x^3 - (n^2 - 1)*x^4)*P[n - 1, x] + (n*x + 4*n*x^2 + 6*n*x^3 + 4*n*x^4 + n*x^5)*D[P[n - 1, x], x])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 + n)*x + (3*n^2 + 3*n)*x^2 + (3*n^2 + 3*n)*x^3 + (n^2 + n)*x^4)*v[i - 2] + ((n + 1)*x - (n^2 - 4*n - 5)*x^2 - (4*n^2 - 6*n - 10)*x^3 - (6*n^2 - 4*n - 10)*x^4 - (4*n^2 - n - 5)*x^5 - (n^2 - 1)*x^6)*deriv(v[i - 2], x) - (n + 1 - (n^2 - 3*n - 4)*x - (3*n^2 - 3*n - 6)*x^2 - (3*n^2 - n - 4)*x^3 - (n^2 - 1)*x^4)*v[i - 1] + (n*x + 4*n*x^2 + 6*n*x^3 + 4*n*x^4 + n*x^5)*deriv(v[i - 1], x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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. [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.