Real-rooted atlas

A059473 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n x + 12\,n x^{2} + 2\,n x^{3} + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(48 + 132\,x + 128\,x^{2} + 52\,x^{3} + 8\,x^{4} - 48\,n - 48\,n x + 34\,n x^{2} + 48\,n x^{3} + 12\,n x^{4} - 36\,n^{2} x - 36\,n^{2} x^{2} + 4\,n^{2} x^{4} - 6\,n^{3} x^{2} - 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(48 + 168\,x + 224\,x^{2} + 140\,x^{3} + 40\,x^{4} + 4\,x^{5} + 48\,n x + 132\,n x^{2} + 128\,n x^{3} + 50\,n x^{4} + 6\,n x^{5} + 12\,n^{2} x^{2} + 24\,n^{2} x^{3} + 14\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
218361266
38024016832520
435614241524608803992
515847920114247008192019230048
67048422887677663776268805568448222784
731360219520480576504320288320917761523210241632128

All 15 rows

Technical details
Node id
n-72f6ffe13aff4619 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-214e6dad78a426b3

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,       4;
  2:     18,      36,       12;
  3:     80,     240,      168,       32;
  4:    356,    1424,     1524,      608,       80;
  5:   1584,    7920,    11424,     7008,     1920,      192;
  6:   7048,   42288,    76776,    63776,    26880,     5568,     448;
  7:  31360,  219520,   480576,   504320,   288320,    91776,   15232,   1024;
  8: 139536, 1116288,  2861280,  3631104,  2627280,  1134720,  288960,  39936,   2304;
  9: 620864, 5587776, 16413696, 24444672, 21424320, 11719872, 4048128, 857088, 101376, 5120;
%F Formula
The row polynomials satisfy (12*n*x+(6*n^2+12*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(48*n-48+(36*n^2+48*n-132)*x+(6*n^3+36*n^2-34*n-128)*x^2+(4*n^3-48*n-52)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (48+(48*n+168)*x+(12*n^2+132*n+224)*x^2+(24*n^2+128*n+140)*x^3+(14*n^2+50*n+40)*x^4+(2*n^2+6*n+4)*x^5)*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[(-(48*n - 48 + (36*n^2 + 48*n - 132)*x + (6*n^3 + 36*n^2 - 34*n - 128)*x^2 + (4*n^3 - 48*n - 52)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (48 + (48*n + 168)*x + (12*n^2 + 132*n + 224)*x^2 + (24*n^2 + 128*n + 140)*x^3 + (14*n^2 + 50*n + 40)*x^4 + (2*n^2 + 6*n + 4)*x^5)*D[P[n - 1, x], x])/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];
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] = (-(48*n - 48 + (36*n^2 + 48*n - 132)*x + (6*n^3 + 36*n^2 - 34*n - 128)*x^2 + (4*n^3 - 48*n - 52)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (48 + (48*n + 168)*x + (12*n^2 + 132*n + 224)*x^2 + (24*n^2 + 128*n + 140)*x^3 + (14*n^2 + 50*n + 40)*x^4 + (2*n^2 + 6*n + 4)*x^5)*deriv(v[i - 1], x))/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)); 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.