Real-rooted atlas

A089231 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 3\,x + 2\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 3\,x + 4\,n + 15\,n x + 13\,n x^{2} + n x^{3} + 5\,n^{2} x + 11\,n^{2} x^{2} + 5\,n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 5\,x - 8\,x^{2} - 5\,x^{3} - x^{4} - 2\,n x - 7\,n x^{2} - 8\,n x^{3} - 3\,n x^{4} - n^{2} x^{2} - 2\,n^{2} x^{3} - n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
213361867
37331232496805
450129204680288060011581
54051300606570062400270004320193531
6376333402849468901226400819000272160352803677647
7394353421489614291928235670402146200011007360296352032256078223657

All 15 rows

Technical details
Node id
n-48447a2879d0dd14 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-66f70a0cb8d78fee

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:        3,         4;
  2:       13,        36,         18;
  3:       73,       312,        324,         96;
  4:      501,      2920,       4680,       2880,         600;
  5:     4051,     30060,      65700,      62400,       27000,        4320;
  6:    37633,    340284,     946890,    1226400,      819000,      272160,      35280;
  7:   394353,   4214896,   14291928,   23567040,    21462000,    11007360,    2963520,     322560;
  8:  4596553,  56786832,  227604384,  457341696,   530258400,   370863360,  154103040,   34836480,   3265920;
  9: 58941091, 827379540, 3833111160, 9104175360, 12862735200, 11453581440, 6490108800, 2264371200, 440899200, 36288000;
%F Formula
The row polynomials satisfy (2+(2*n+3)*x+(n^2+n)*x^2)*P(n,x) = (4*n+2+(5*n^2+15*n+3)*x+(2*n^3+11*n^2+13*n)*x^2+(2*n^3+5*n^2+n)*x^3)*P(n-1,x) - (1+(2*n+5)*x+(n^2+7*n+8)*x^2+(2*n^2+8*n+5)*x^3+(n^2+3*n+1)*x^4)*d/dx 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[Together[((4*n + 2 + (5*n^2 + 15*n + 3)*x + (2*n^3 + 11*n^2 + 13*n)*x^2 + (2*n^3 + 5*n^2 + n)*x^3)*P[n - 1, x] - (1 + (2*n + 5)*x + (n^2 + 7*n + 8)*x^2 + (2*n^2 + 8*n + 5)*x^3 + (n^2 + 3*n + 1)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 3)*x + (n^2 + n)*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] = ((4*n + 2 + (5*n^2 + 15*n + 3)*x + (2*n^3 + 11*n^2 + 13*n)*x^2 + (2*n^3 + 5*n^2 + n)*x^3)*v[i - 1] - (1 + (2*n + 5)*x + (n^2 + 7*n + 8)*x^2 + (2*n^2 + 8*n + 5)*x^3 + (n^2 + 3*n + 1)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 3)*x + (n^2 + n)*x^2)); 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. [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.