Real-rooted atlas

A099040 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5\,n - 14\,n x - 4\,n x^{2} + 8\,n x^{3} + 8\,n^{2} x + 8\,n^{2} x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-24 - 68\,x - 24\,x^{2} + 48\,x^{3} + 32\,x^{4} - 14\,n + 12\,n x + 40\,n x^{2} + 16\,n x^{3} + 24\,n^{2} + 8\,n^{2} x - 16\,n^{2} x^{2} - 4\,n^{3}\right)\,P_{n-1}(x) + \left(-3 - 2\,x + 48\,x^{2} + 128\,x^{3} + 112\,x^{4} + 32\,x^{5} + 24\,n x + 88\,n x^{2} + 96\,n x^{3} + 32\,n x^{4} + 6\,n^{2} + 16\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
226521694
39633632064816
43321664280017282566780
51096718417504191368704102454648
635122846491168144960116736419844096430920
7110081063684229128867841040384661504196608163843341952

All 15 rows

Technical details
Node id
n-dfe62937de87f91f · 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 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:      6,       4;
  2:     26,      52,      16;
  3:     96,     336,     320,       64;
  4:    332,    1664,    2800,     1728,      256;
  5:   1096,    7184,   17504,    19136,     8704,     1024;
  6:   3512,   28464,   91168,   144960,   116736,    41984,     4096;
  7:  11008,  106368,  422912,   886784,  1040384,   661504,   196608,    16384;
  8:  33936,  380928, 1810560,  4724224,  7321856,  6785024,  3559424,   901120,   65536;
  9: 103264, 1320768, 7307520, 22836736, 44109312, 54043648, 41312256, 18432000, 4063232, 262144;
%F Formula
The row polynomials satisfy (2*n^3-5*n+(8*n^2-14*n)*x+(8*n^2-4*n)*x^2+8*n*x^3)*P(n,x) = -(4*n^3-24*n^2+14*n+24-(8*n^2+12*n-68)*x+(16*n^2-40*n+24)*x^2-(16*n+48)*x^3-32*x^4)*P(n-1,x) + (6*n^2-3+(16*n^2+24*n-2)*x+(8*n^2+88*n+48)*x^2+(96*n+128)*x^3+(32*n+112)*x^4+32*x^5)*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[(-(4*n^3 - 24*n^2 + 14*n + 24 - (8*n^2 + 12*n - 68)*x + (16*n^2 - 40*n + 24)*x^2 - (16*n + 48)*x^3 - 32*x^4)*P[n - 1, x] + (6*n^2 - 3 + (16*n^2 + 24*n - 2)*x + (8*n^2 + 88*n + 48)*x^2 + (96*n + 128)*x^3 + (32*n + 112)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(2*n^3 - 5*n + (8*n^2 - 14*n)*x + (8*n^2 - 4*n)*x^2 + 8*n*x^3)]];
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] = (-(4*n^3 - 24*n^2 + 14*n + 24 - (8*n^2 + 12*n - 68)*x + (16*n^2 - 40*n + 24)*x^2 - (16*n + 48)*x^3 - 32*x^4)*v[i - 1] + (6*n^2 - 3 + (16*n^2 + 24*n - 2)*x + (8*n^2 + 88*n + 48)*x^2 + (96*n + 128)*x^3 + (32*n + 112)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(2*n^3 - 5*n + (8*n^2 - 14*n)*x + (8*n^2 - 4*n)*x^2 + 8*n*x^3)); 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

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.