Real-rooted atlas

A290596 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(14 + 107\,x + 225\,x^{2} + 186\,x^{3} + 54\,x^{4} - 6\,n - 69\,n x - 165\,n x^{2} - 147\,n x^{3} - 45\,n x^{4} + 9\,n^{2} x + 27\,n^{2} x^{2} + 27\,n^{2} x^{3} + 9\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(6 + 84\,x + 270\,x^{2} + 366\,x^{3} + 228\,x^{4} + 54\,x^{5} - 18\,n x - 72\,n x^{2} - 108\,n x^{3} - 72\,n x^{4} - 18\,n x^{5}\right)\,P_{n-1}'(x) + \left(9\,x + 45\,x^{2} + 90\,x^{3} + 90\,x^{4} + 45\,x^{5} + 9\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
310302161
4803605042251169
58805280110889900321330361
6123209240025872034650022491057051991901
72094401884960659736011781000114704105819202120795338970325
8418880043982400184726080412335000535285800407344140169113420296525611786628201

All 15 rows

Technical details
Node id
n-33b9112ff4543bf3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5c89fc9d5ddc5573

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:          2,           3;
   3:         10,          30,           21;
   4:         80,         360,          504,          225;
   5:        880,        5280,        11088,         9900,          3213;
   6:      12320,       92400,       258720,       346500,        224910,         57051;
   7:     209440,     1884960,      6597360,     11781000,      11470410,       5819202,       1207953;
   8:    4188800,    43982400,    184726080,    412335000,     535285800,     407344140,     169113420,     29652561;
   9:   96342400,  1156108800,   5664933120,  15173928000,   24623146800,   24983773920,   15558434640,   5456071224,    827024985;
  10: 2504902400, 33816182400, 189370621440, 591783192000, 1152363270240, 1461550774320, 1213557901920, 638360333208, 193523846490, 25810060563;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(6*n-14-(9*n^2-69*n+107)*x-(27*n^2-165*n+225)*x^2-(27*n^2-147*n+186)*x^3-(9*n^2-45*n+54)*x^4)*P(n-1,x) + (6-(18*n-84)*x-(72*n-270)*x^2-(108*n-366)*x^3-(72*n-228)*x^4-(18*n-54)*x^5)*d/dx P(n-1,x) + (9*x+45*x^2+90*x^3+90*x^4+45*x^5+9*x^6)*d^2/dx^2 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[Together[(-(6*n - 14 - (9*n^2 - 69*n + 107)*x - (27*n^2 - 165*n + 225)*x^2 - (27*n^2 - 147*n + 186)*x^3 - (9*n^2 - 45*n + 54)*x^4)*P[n - 1, x] + (6 - (18*n - 84)*x - (72*n - 270)*x^2 - (108*n - 366)*x^3 - (72*n - 228)*x^4 - (18*n - 54)*x^5)*D[P[n - 1, x], x] + (9*x + 45*x^2 + 90*x^3 + 90*x^4 + 45*x^5 + 9*x^6)*D[P[n - 1, x], {x, 2}])/(1 + 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] = (-(6*n - 14 - (9*n^2 - 69*n + 107)*x - (27*n^2 - 165*n + 225)*x^2 - (27*n^2 - 147*n + 186)*x^3 - (9*n^2 - 45*n + 54)*x^4)*v[i - 1] + (6 - (18*n - 84)*x - (72*n - 270)*x^2 - (108*n - 366)*x^3 - (72*n - 228)*x^4 - (18*n - 54)*x^5)*deriv(v[i - 1], x) + (9*x + 45*x^2 + 90*x^3 + 90*x^4 + 45*x^5 + 9*x^6)*deriv(deriv(v[i - 1], x), x))/(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

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.