Real-rooted atlas

A113953 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n + 23\,n x + 15\,n x^{2} + 9\,n^{2} x + 21\,n^{2} x^{2} + 6\,n^{3} x^{2}\right)\,P_{n}(x) = \left(30 + 142\,x + 186\,x^{2} + 90\,x^{3} - 24\,n - 20\,n x + 126\,n x^{2} + 126\,n x^{3} - 54\,n^{2} x - 54\,n^{2} x^{2} + 36\,n^{2} x^{3} - 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(9 + 72\,x + 194\,x^{2} + 192\,x^{3} + 45\,x^{4} + 27\,n x + 153\,n x^{2} + 237\,n x^{3} + 63\,n x^{4} + 18\,n^{2} x^{2} + 60\,n^{2} x^{3} + 18\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1167
23242754
3566189108368
41118075610804052432
521438253858325265145815552
64310567641241923604523328510397408
785244221411867241810351939149695717496600064

All 15 rows

Technical details
Node id
n-2abd14fc158c8140 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1920365dcfdf3eae

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:   1,     6;
  2:   3,    24,     27;
  3:   5,    66,    189,    108;
  4:  11,   180,    756,   1080,     405;
  5:  21,   438,   2538,   5832,    5265,    1458;
  6:  43,  1056,   7641,  24192,   36045,   23328,    5103;
  7:  85,  2442,  21411,  86724,  181035,  193914,   96957,   17496;
  8: 171,  5580,  57186, 281880,  759780, 1151820,  949158,  384912,   59049;
  9: 341, 12510, 147096, 855360, 2829330, 5578308, 6531840, 4339008, 1476225, 196830;
%F Formula
The row polynomials satisfy (6*n+(9*n^2+23*n)*x+(6*n^3+21*n^2+15*n)*x^2)*P(n,x) = -(24*n-30+(54*n^2+20*n-142)*x+(24*n^3+54*n^2-126*n-186)*x^2-(36*n^2+126*n+90)*x^3)*P(n-1,x) + (9+(27*n+72)*x+(18*n^2+153*n+194)*x^2+(60*n^2+237*n+192)*x^3+(18*n^2+63*n+45)*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[(-(24*n - 30 + (54*n^2 + 20*n - 142)*x + (24*n^3 + 54*n^2 - 126*n - 186)*x^2 - (36*n^2 + 126*n + 90)*x^3)*P[n - 1, x] + (9 + (27*n + 72)*x + (18*n^2 + 153*n + 194)*x^2 + (60*n^2 + 237*n + 192)*x^3 + (18*n^2 + 63*n + 45)*x^4)*D[P[n - 1, x], x])/(6*n + (9*n^2 + 23*n)*x + (6*n^3 + 21*n^2 + 15*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] = (-(24*n - 30 + (54*n^2 + 20*n - 142)*x + (24*n^3 + 54*n^2 - 126*n - 186)*x^2 - (36*n^2 + 126*n + 90)*x^3)*v[i - 1] + (9 + (27*n + 72)*x + (18*n^2 + 153*n + 194)*x^2 + (60*n^2 + 237*n + 192)*x^3 + (18*n^2 + 63*n + 45)*x^4)*deriv(v[i - 1], x))/(6*n + (9*n^2 + 23*n)*x + (6*n^3 + 21*n^2 + 15*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.