Real-rooted atlas

A063967 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 22\,x + 12\,x^{2} - 8\,x^{3} + n - 30\,n x - 20\,n x^{2} + 8\,n x^{3} - 6\,n^{2} + 8\,n^{2} x + 8\,n^{2} x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(18 - 72\,x - 80\,x^{2} + 32\,x^{3} + 32\,x^{4} - 74\,n - 4\,n x + 72\,n x^{2} + 16\,n x^{3} + 36\,n^{2} + 8\,n^{2} x - 16\,n^{2} x^{2} - 4\,n^{3}\right)\,P_{n-1}(x) + \left(3 - 10\,x - 32\,x^{2} + 32\,x^{3} + 80\,x^{4} + 32\,x^{5} - 12\,n - 8\,n x + 72\,n x^{2} + 96\,n x^{3} + 32\,n x^{4} + 6\,n^{2} + 16\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
326521694
49633632064816
53321664280017282566780
61096718417504191368704102454648
735122846491168144960116736419844096430920
8110081063684229128867841040384661504196608163843341952

All 15 rows

Technical details
Node id
n-09ca1d900013a9af · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-10b1f054401300a3

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:      6,       4;
   3:     26,      52,      16;
   4:     96,     336,     320,       64;
   5:    332,    1664,    2800,     1728,      256;
   6:   1096,    7184,   17504,    19136,     8704,     1024;
   7:   3512,   28464,   91168,   144960,   116736,    41984,     4096;
   8:  11008,  106368,  422912,   886784,  1040384,   661504,   196608,    16384;
   9:  33936,  380928, 1810560,  4724224,  7321856,  6785024,  3559424,   901120,   65536;
  10: 103264, 1320768, 7307520, 22836736, 44109312, 54043648, 41312256, 18432000, 4063232, 262144;
%F Formula
The row polynomials satisfy (2*n^3-6*n^2+n+3+(8*n^2-30*n+22)*x+(8*n^2-20*n+12)*x^2+(8*n-8)*x^3)*P(n,x) = -(4*n^3-36*n^2+74*n-18-(8*n^2-4*n-72)*x+(16*n^2-72*n+80)*x^2-(16*n+32)*x^3-32*x^4)*P(n-1,x) + (6*n^2-12*n+3+(16*n^2-8*n-10)*x+(8*n^2+72*n-32)*x^2+(96*n+32)*x^3+(32*n+80)*x^4+32*x^5)*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[(-(4*n^3 - 36*n^2 + 74*n - 18 - (8*n^2 - 4*n - 72)*x + (16*n^2 - 72*n + 80)*x^2 - (16*n + 32)*x^3 - 32*x^4)*P[n - 1, x] + (6*n^2 - 12*n + 3 + (16*n^2 - 8*n - 10)*x + (8*n^2 + 72*n - 32)*x^2 + (96*n + 32)*x^3 + (32*n + 80)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + n + 3 + (8*n^2 - 30*n + 22)*x + (8*n^2 - 20*n + 12)*x^2 + (8*n - 8)*x^3)]];
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] = (-(4*n^3 - 36*n^2 + 74*n - 18 - (8*n^2 - 4*n - 72)*x + (16*n^2 - 72*n + 80)*x^2 - (16*n + 32)*x^3 - 32*x^4)*v[i - 1] + (6*n^2 - 12*n + 3 + (16*n^2 - 8*n - 10)*x + (8*n^2 + 72*n - 32)*x^2 + (96*n + 32)*x^3 + (32*n + 80)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + n + 3 + (8*n^2 - 30*n + 22)*x + (8*n^2 - 20*n + 12)*x^2 + (8*n - 8)*x^3)); 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

Same fingerprint ?

References

  1. [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
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.