Real-rooted atlas

A099093 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(14\,n + 19\,n x + 5\,n x^{2} + 7\,n^{2} x + 6\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(70 + 158\,x + 118\,x^{2} + 30\,x^{3} - 28\,n + 25\,n x + 83\,n x^{2} + 36\,n x^{3} - 21\,n^{2} x - 7\,n^{2} x^{2} + 6\,n^{2} x^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(49 + 154\,x + 176\,x^{2} + 86\,x^{3} + 15\,x^{4} + 49\,n x + 112\,n x^{2} + 81\,n x^{3} + 18\,n x^{4} + 7\,n^{2} x^{2} + 10\,n^{2} x^{3} + 3\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
212422781
345234297108684
41711170221416204055580
5648550813851152287695145844388
62457249487824611480487480335345103346572
79315109998413343758484773145447606137781174962667168
835316475470208202445862205856705454167021126425423765904920291472
91338932024190101207072601558039932190388352882421373593953522066715196830152934480
105076278514234478566631406163242521234352949708962307984841200750483991712476763706495391143705744
1119245603546439222135501873180519215010365152052818802193335830112627388085629141171635657302793017721257648497037376
1272965611465290001005538860369411127285310605351335474491414709734298115684076886475490487252316377065083807899910908690873362773735104
1327663363601306902449965189818185324904466934778908233824052810385575080395555471256644203332363152153718010902512115252753339635234538322320522461503469376
14104879772245326725019880833833876590715602477215009354858872946566906219043507278818714645742305626053384535152601470075637144569329718096169563361227628710717445353378511892160
Technical details
Node id
n-a8738605bf0751ad · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5c2945226871807e

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,       6;
  2:     12,      42,       27;
  3:     45,     234,      297,      108;
  4:    171,    1170,     2214,     1620,      405;
  5:    648,    5508,    13851,    15228,     7695,     1458;
  6:   2457,   24948,    78246,   114804,    87480,    33534,     5103;
  7:   9315,  109998,   413343,   758484,   773145,   447606,   137781,   17496;
  8:  35316,  475470,  2082024,  4586220,  5856705,  4541670,  2112642,  542376,   59049;
  9: 133893, 2024190, 10120707, 26015580, 39932190, 38835288, 24213735, 9395352, 2066715, 196830;
%F Formula
The row polynomials satisfy (14*n+(7*n^2+19*n)*x+(n^3+6*n^2+5*n)*x^2)*P(n,x) = -(28*n-70+(21*n^2-25*n-158)*x+(2*n^3+7*n^2-83*n-118)*x^2-(6*n^2+36*n+30)*x^3)*P(n-1,x) + (49+(49*n+154)*x+(7*n^2+112*n+176)*x^2+(10*n^2+81*n+86)*x^3+(3*n^2+18*n+15)*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[(-(28*n - 70 + (21*n^2 - 25*n - 158)*x + (2*n^3 + 7*n^2 - 83*n - 118)*x^2 - (6*n^2 + 36*n + 30)*x^3)*P[n - 1, x] + (49 + (49*n + 154)*x + (7*n^2 + 112*n + 176)*x^2 + (10*n^2 + 81*n + 86)*x^3 + (3*n^2 + 18*n + 15)*x^4)*D[P[n - 1, x], x])/(14*n + (7*n^2 + 19*n)*x + (n^3 + 6*n^2 + 5*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] = (-(28*n - 70 + (21*n^2 - 25*n - 158)*x + (2*n^3 + 7*n^2 - 83*n - 118)*x^2 - (6*n^2 + 36*n + 30)*x^3)*v[i - 1] + (49 + (49*n + 154)*x + (7*n^2 + 112*n + 176)*x^2 + (10*n^2 + 81*n + 86)*x^3 + (3*n^2 + 18*n + 15)*x^4)*deriv(v[i - 1], x))/(14*n + (7*n^2 + 19*n)*x + (n^3 + 6*n^2 + 5*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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.