Real-rooted atlas

A261711 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-48\,n + 20\,n x + 24\,n^{2} x + 2\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-24 + 46\,x - 20\,x^{2} - 2\,x^{3} - 168\,n + 52\,n x - 3\,n x^{2} - n x^{3} + 78\,n^{2} x + 32\,n^{2} x^{2} + 2\,n^{2} x^{3} + 7\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-72 + 108\,x - 4\,x^{2} - 28\,x^{3} - 4\,x^{4} + 72\,n x - 30\,n x^{2} - 36\,n x^{3} - 6\,n x^{4} + 6\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
144
215217
3561672
4209923304
5780464361280
62911218227945376
710864982417766422528
8405454293610098624594208
9151316183648533524800100393216
1056471977274626771432040117561638400
11210756032099681292976194496106001446815744
1278655211319656460633091102896812751980728311552
13293545245379140827775212594105655762020496196117440512
1410955257521770011012483238530745740352782517753430878486539264
Technical details
Node id
n-3113468fc7a7b150 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bfb083e0f5777494

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      4;
  2:     15,      2;
  3:     56,     16;
  4:    209,     92,     3;
  5:    780,    464,    36;
  6:   2911,   2182,   279,    4;
  7:  10864,   9824,  1776,   64;
  8:  40545,  42936, 10098,  624,   5;
  9: 151316, 183648, 53352, 4800, 100;
%F Formula
The row polynomials satisfy -(48*n-(24*n^2+20*n)*x-(2*n^3+2*n^2)*x^2)*P(n,x) = -(168*n+24-(78*n^2+52*n+46)*x-(7*n^3+32*n^2-3*n-20)*x^2-(n^3+2*n^2-n-2)*x^3)*P(n-1,x) - (72-(72*n+108)*x-(6*n^2-30*n-4)*x^2+(4*n^2+36*n+28)*x^3+(2*n^2+6*n+4)*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[(-(168*n + 24 - (78*n^2 + 52*n + 46)*x - (7*n^3 + 32*n^2 - 3*n - 20)*x^2 - (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] - (72 - (72*n + 108)*x - (6*n^2 - 30*n - 4)*x^2 + (4*n^2 + 36*n + 28)*x^3 + (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(-48*n + (24*n^2 + 20*n)*x + (2*n^3 + 2*n^2)*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] = (-(168*n + 24 - (78*n^2 + 52*n + 46)*x - (7*n^3 + 32*n^2 - 3*n - 20)*x^2 - (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] - (72 - (72*n + 108)*x - (6*n^2 - 30*n - 4)*x^2 + (4*n^2 + 36*n + 28)*x^3 + (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(-48*n + (24*n^2 + 20*n)*x + (2*n^3 + 2*n^2)*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. [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.