Real-rooted atlas

A063967 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-48 - 20\,x + 48\,n + 8\,n x - 3\,n x^{2} + 12\,n^{2} x + 2\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(288 + 144\,x + 16\,x^{2} - 96\,n + 92\,n x + 54\,n x^{2} + 6\,n x^{3} - 36\,n^{2} x + 2\,n^{2} x^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(288 + 288\,x + 88\,x^{2} + 8\,x^{3} + 144\,n x + 132\,n x^{2} + 36\,n x^{3} + 3\,n x^{4} + 12\,n^{2} x^{2} + 8\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3610319
4164024484
544144129445366
61204885823007061566
732815922370163658010276615
88965056902478083720996140827648

All 15 rows

Technical details
Node id
n-9ae74366a6508bea · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f712355985a498aa

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,     2;
   3:    6,    10,      3;
   4:   16,    40,     24,      4;
   5:   44,   144,    129,     44,      5;
   6:  120,   488,    582,    300,     70,     6;
   7:  328,  1592,   2370,   1636,    580,   102,     7;
   8:  896,  5056,   9024,   7808,   3720,   996,   140,    8;
   9: 2448, 15744,  32760,  34080,  20405,  7356,  1575,  184,   9;
  10: 6688, 48288, 114768, 139488, 100610, 45654, 13174, 2344, 234, 10;
%F Formula
The row polynomials satisfy (48*n-48+(12*n^2+8*n-20)*x+(n^3+2*n^2-3*n)*x^2)*P(n,x) = -(96*n-288+(36*n^2-92*n-144)*x+(2*n^3-54*n-16)*x^2-(2*n^2+6*n)*x^3)*P(n-1,x) + (288+(144*n+288)*x+(12*n^2+132*n+88)*x^2+(8*n^2+36*n+8)*x^3+(n^2+3*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[(-(96*n - 288 + (36*n^2 - 92*n - 144)*x + (2*n^3 - 54*n - 16)*x^2 - (2*n^2 + 6*n)*x^3)*P[n - 1, x] + (288 + (144*n + 288)*x + (12*n^2 + 132*n + 88)*x^2 + (8*n^2 + 36*n + 8)*x^3 + (n^2 + 3*n)*x^4)*D[P[n - 1, x], x])/(48*n - 48 + (12*n^2 + 8*n - 20)*x + (n^3 + 2*n^2 - 3*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] = (-(96*n - 288 + (36*n^2 - 92*n - 144)*x + (2*n^3 - 54*n - 16)*x^2 - (2*n^2 + 6*n)*x^3)*v[i - 1] + (288 + (144*n + 288)*x + (12*n^2 + 132*n + 88)*x^2 + (8*n^2 + 36*n + 8)*x^3 + (n^2 + 3*n)*x^4)*deriv(v[i - 1], x))/(48*n - 48 + (12*n^2 + 8*n - 20)*x + (n^3 + 2*n^2 - 3*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.