Real-rooted atlas

A099097 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-78 - 72\,x + 78\,n + 33\,n x - 30\,n x^{2} + 39\,n^{2} x + 27\,n^{2} x^{2} + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(338 + 520\,x + 198\,x^{2} - 52\,n + 277\,n x + 481\,n x^{2} + 180\,n x^{3} - 39\,n^{2} x - 7\,n^{2} x^{2} + 18\,n^{2} x^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(169 + 429\,x + 359\,x^{2} + 99\,x^{3} + 169\,n x + 416\,n x^{2} + 337\,n x^{3} + 90\,n x^{4} + 13\,n^{2} x^{2} + 22\,n^{2} x^{3} + 9\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2369
310382775
433186261108588
5109814170414044054436
636033489315114846615145832580
7118913236459367538865070286745103234596
8392750910211707433812498885329670117369174961663776

All 15 rows

Technical details
Node id
n-96a7ee81bc091ff1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-866a1dcf97397fb3

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:     3,      6;
   3:    10,     38,      27;
   4:    33,    186,     261,      108;
   5:   109,    814,    1704,     1404,      405;
   6:   360,   3348,    9315,    11484,     6615,     1458;
   7:  1189,  13236,   45936,    75388,    65070,    28674,     5103;
   8:  3927,  50910,  211707,   433812,   498885,   329670,   117369,   17496;
   9: 12970, 191906,  930006,  2284900,  3282755,  2891754,  1544508,  460728,   59049;
  10: 42837, 712302, 3941667, 11290812, 19457790, 21421512, 15256647, 6829272, 1751787, 196830;
%F Formula
The row polynomials satisfy (78*n-78+(39*n^2+33*n-72)*x+(3*n^3+27*n^2-30*n)*x^2)*P(n,x) = -(52*n-338+(39*n^2-277*n-520)*x+(2*n^3+7*n^2-481*n-198)*x^2-(18*n^2+180*n)*x^3)*P(n-1,x) + (169+(169*n+429)*x+(13*n^2+416*n+359)*x^2+(22*n^2+337*n+99)*x^3+(9*n^2+90*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(52*n - 338 + (39*n^2 - 277*n - 520)*x + (2*n^3 + 7*n^2 - 481*n - 198)*x^2 - (18*n^2 + 180*n)*x^3)*P[n - 1, x] + (169 + (169*n + 429)*x + (13*n^2 + 416*n + 359)*x^2 + (22*n^2 + 337*n + 99)*x^3 + (9*n^2 + 90*n)*x^4)*D[P[n - 1, x], x])/(78*n - 78 + (39*n^2 + 33*n - 72)*x + (3*n^3 + 27*n^2 - 30*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(52*n - 338 + (39*n^2 - 277*n - 520)*x + (2*n^3 + 7*n^2 - 481*n - 198)*x^2 - (18*n^2 + 180*n)*x^3)*v[i - 1] + (169 + (169*n + 429)*x + (13*n^2 + 416*n + 359)*x^2 + (22*n^2 + 337*n + 99)*x^3 + (9*n^2 + 90*n)*x^4)*deriv(v[i - 1], x))/(78*n - 78 + (39*n^2 + 33*n - 72)*x + (3*n^3 + 27*n^2 - 30*n)*x^2)); 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

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.