Real-rooted atlas

A008278 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-x - x^{3} + n x^{3}\right)\,P_{n}(x) = \left(-2 - 7\,x - 4\,x^{2} + 2\,x^{3} + 2\,n + 3\,n x + 4\,n x^{3} + 4\,n x^{4} - 2\,n x^{5} + n^{2} x - 2\,n^{2} x^{3} + 2\,n^{2} x^{4} + 3\,n^{2} x^{5} - n^{3} x^{3} - 2\,n^{3} x^{4} - n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-1 - 3\,x - 2\,x^{2} + x^{3} + x^{4} - n x - 2\,n x^{2} + n x^{4} - n x^{5} - n x^{6} + n^{2} x^{3} + 3\,n^{2} x^{4} + 3\,n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-2 - x + 2\,x^{2} + 2\,x^{4} - 2\,n x - 4\,n x^{4} + 2\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + 2\,x^{3} + 2\,x^{4} + x^{5} - n x^{3} - 2\,n x^{4} - n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3510318
4154630495
552218228765579
62031088159389618063966
787757521092388163155414729944
8414032226756248035642355104949388246141

All 15 rows

Technical details
Node id
n-ab40c2db612c2433 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5f72f3d1614d5d16

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,       2;
   3:      5,      10,       3;
   4:     15,      46,      30,       4;
   5:     52,     218,     228,      76,       5;
   6:    203,    1088,    1593,     896,     180,       6;
   7:    877,    5752,   10923,    8816,    3155,     414,      7;
   8:   4140,   32226,   75624,   80356,   42355,   10494,    938,      8;
   9:  21147,  190990,  535692,  710524,  500855,  187986,  33866,   2104,    9;
  10: 115975, 1194310, 3907353, 6239152, 5547365, 2830962, 796649, 107632, 4680, 10;
%F Formula
The row polynomials satisfy -(x-(n-1)*x^3)*P(n,x) = (2*n-2+(n^2+3*n-7)*x-4*x^2-(n^3+2*n^2-4*n-2)*x^3-(2*n^3-2*n^2-4*n)*x^4-(n^3-3*n^2+2*n)*x^5)*P(n-2,x) - (1+(n+3)*x+(2*n+2)*x^2-(n^2+1)*x^3-(3*n^2+n+1)*x^4-(3*n^2-n)*x^5-(n^2-n)*x^6)*d/dx P(n-2,x) - (2+(2*n+1)*x-2*x^2-2*n^2*x^3-(2*n^2-4*n+2)*x^4)*P(n-1,x) + (x+2*x^2-(n-2)*x^3-(2*n-2)*x^4-(n-1)*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n^2 + 3*n - 7)*x - 4*x^2 - (n^3 + 2*n^2 - 4*n - 2)*x^3 - (2*n^3 - 2*n^2 - 4*n)*x^4 - (n^3 - 3*n^2 + 2*n)*x^5)*P[n - 2, x] - (1 + (n + 3)*x + (2*n + 2)*x^2 - (n^2 + 1)*x^3 - (3*n^2 + n + 1)*x^4 - (3*n^2 - n)*x^5 - (n^2 - n)*x^6)*D[P[n - 2, x], x] - (2 + (2*n + 1)*x - 2*x^2 - 2*n^2*x^3 - (2*n^2 - 4*n + 2)*x^4)*P[n - 1, x] + (x + 2*x^2 - (n - 2)*x^3 - (2*n - 2)*x^4 - (n - 1)*x^5)*D[P[n - 1, x], x])/(-x + (n - 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = ((2*n - 2 + (n^2 + 3*n - 7)*x - 4*x^2 - (n^3 + 2*n^2 - 4*n - 2)*x^3 - (2*n^3 - 2*n^2 - 4*n)*x^4 - (n^3 - 3*n^2 + 2*n)*x^5)*v[i - 2] - (1 + (n + 3)*x + (2*n + 2)*x^2 - (n^2 + 1)*x^3 - (3*n^2 + n + 1)*x^4 - (3*n^2 - n)*x^5 - (n^2 - n)*x^6)*deriv(v[i - 2], x) - (2 + (2*n + 1)*x - 2*x^2 - 2*n^2*x^3 - (2*n^2 - 4*n + 2)*x^4)*v[i - 1] + (x + 2*x^2 - (n - 2)*x^3 - (2*n - 2)*x^4 - (n - 1)*x^5)*deriv(v[i - 1], x))/(-x + (n - 1)*x^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. [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.