Real-rooted atlas

A063967 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-16 - 20\,x + 16\,n + 8\,n x - 9\,n x^{2} + 12\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(96 + 144\,x + 48\,x^{2} - 32\,n + 92\,n x + 162\,n x^{2} + 54\,n x^{3} - 36\,n^{2} x + 18\,n^{2} x^{3} - 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(32 + 96\,x + 88\,x^{2} + 24\,x^{3} + 48\,n x + 132\,n x^{2} + 108\,n x^{3} + 27\,n x^{4} + 12\,n^{2} x^{2} + 24\,n^{2} x^{3} + 9\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2268
36302763
416120216108460
544432116111884053230
61201464523881005670145822050
73284776213304417246980247865103147475
8896151688121621081630132024202810206017496971000

All 15 rows

Technical details
Node id
n-f99339a05019c08b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-93ca8a21ebaf7349

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:    2,      6;
   3:    6,     30,      27;
   4:   16,    120,     216,     108;
   5:   44,    432,    1161,    1188,     405;
   6:  120,   1464,    5238,    8100,    5670,     1458;
   7:  328,   4776,   21330,   44172,   46980,    24786,    5103;
   8:  896,  15168,   81216,  210816,  301320,   242028,  102060,   17496;
   9: 2448,  47232,  294840,  920160, 1652805,  1787508, 1148175,  402408,   59049;
  10: 6688, 144864, 1032912, 3766176, 8149410, 11093922, 9603846, 5126328, 1535274, 196830;
%F Formula
The row polynomials satisfy (16*n-16+(12*n^2+8*n-20)*x+(3*n^3+6*n^2-9*n)*x^2)*P(n,x) = -(32*n-96+(36*n^2-92*n-144)*x+(6*n^3-162*n-48)*x^2-(18*n^2+54*n)*x^3)*P(n-1,x) + (32+(48*n+96)*x+(12*n^2+132*n+88)*x^2+(24*n^2+108*n+24)*x^3+(9*n^2+27*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[(-(32*n - 96 + (36*n^2 - 92*n - 144)*x + (6*n^3 - 162*n - 48)*x^2 - (18*n^2 + 54*n)*x^3)*P[n - 1, x] + (32 + (48*n + 96)*x + (12*n^2 + 132*n + 88)*x^2 + (24*n^2 + 108*n + 24)*x^3 + (9*n^2 + 27*n)*x^4)*D[P[n - 1, x], x])/(16*n - 16 + (12*n^2 + 8*n - 20)*x + (3*n^3 + 6*n^2 - 9*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] = (-(32*n - 96 + (36*n^2 - 92*n - 144)*x + (6*n^3 - 162*n - 48)*x^2 - (18*n^2 + 54*n)*x^3)*v[i - 1] + (32 + (48*n + 96)*x + (12*n^2 + 132*n + 88)*x^2 + (24*n^2 + 108*n + 24)*x^3 + (9*n^2 + 27*n)*x^4)*deriv(v[i - 1], x))/(16*n - 16 + (12*n^2 + 8*n - 20)*x + (3*n^3 + 6*n^2 - 9*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.