Real-rooted atlas

A345013 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 6\,n^{2} + 4\,n^{2} x + 2\,n^{3} + 4\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} + 16\,n + 28\,n x + 14\,n x^{2} + 20\,n^{2} + 42\,n^{2} x + 19\,n^{2} x^{2} + 8\,n^{3} + 26\,n^{3} x + 14\,n^{3} x^{2} + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} - 4\,x^{3} - 8\,n x - 18\,n x^{2} - 10\,n x^{3} - 4\,n^{2} x - 13\,n^{2} x^{2} - 9\,n^{2} x^{3} - 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
215401873
35621018040486
421010081260560753113
5792462075605040140012619538
63003205924158036960157503024196121105
71144090090216216240240138600415805880288744334

All 15 rows

Technical details
Node id
n-c83c4c7cfb1526d4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b982193f04b77e58

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:      4,       6;
  2:     15,      40,      18;
  3:     56,     210,     180,      40;
  4:    210,    1008,    1260,     560,      75;
  5:    792,    4620,    7560,    5040,    1400,     126;
  6:   3003,   20592,   41580,   36960,   15750,    3024,     196;
  7:  11440,   90090,  216216,  240240,  138600,   41580,    5880,    288;
  8:  43758,  388960, 1081080, 1441440, 1051050,  432432,   97020,  10560,   405;
  9: 167960, 1662804, 5250960, 8168160, 7207200, 3783780, 1177176, 205920, 17820, 550;
%F Formula
The row polynomials satisfy (2*n^3+6*n^2+4*n+(n^4+4*n^3+4*n^2)*x)*P(n,x) = (8*n^3+20*n^2+16*n+4+(4*n^4+26*n^3+42*n^2+28*n+8)*x+(3*n^4+14*n^3+19*n^2+14*n+4)*x^2)*P(n-1,x) - ((4*n^2+8*n+4)*x+(2*n^3+13*n^2+18*n+8)*x^2+(2*n^3+9*n^2+10*n+4)*x^3)*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[((8*n^3 + 20*n^2 + 16*n + 4 + (4*n^4 + 26*n^3 + 42*n^2 + 28*n + 8)*x + (3*n^4 + 14*n^3 + 19*n^2 + 14*n + 4)*x^2)*P[n - 1, x] - ((4*n^2 + 8*n + 4)*x + (2*n^3 + 13*n^2 + 18*n + 8)*x^2 + (2*n^3 + 9*n^2 + 10*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 6*n^2 + 4*n + (n^4 + 4*n^3 + 4*n^2)*x)]];
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] = ((8*n^3 + 20*n^2 + 16*n + 4 + (4*n^4 + 26*n^3 + 42*n^2 + 28*n + 8)*x + (3*n^4 + 14*n^3 + 19*n^2 + 14*n + 4)*x^2)*v[i - 1] - ((4*n^2 + 8*n + 4)*x + (2*n^3 + 13*n^2 + 18*n + 8)*x^2 + (2*n^3 + 9*n^2 + 10*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 6*n^2 + 4*n + (n^4 + 4*n^3 + 4*n^2)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.