Real-rooted atlas

A052553 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 76\,x - 32\,x^{2} + 14\,n - 108\,n x + 32\,n x^{2} - 28\,n^{2} + 32\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(30 + 268\,x + 192\,x^{2} - 256\,x^{3} + 66\,n - 172\,n x - 336\,n x^{2} + 128\,n x^{3} - 92\,n^{2} - 48\,n^{2} x + 128\,n^{2} x^{2} + 24\,n^{3} + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(5 + 48\,x + 44\,x^{2} - 192\,x^{3} - 256\,x^{4} + 10\,n - 44\,n x - 256\,n x^{2} - 256\,n x^{3} - 20\,n^{2} - 72\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3426
47411
51518437
628441284
753112608233
89725218432565
9176550540176161458
1031511481404640803587
11559233434522120480328977
12984463280406208200019222056
13172190241802416912736012486454353
142993173043912043296238565824448132841
155180327468272810576071120235203136128324318
Technical details
Node id
n-3601965de4644d86 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-594a37631f56a83b

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:   1;
   3:   4,    2;
   4:   7,    4;
   5:  15,   18,    4;
   6:  28,   44,   12;
   7:  53,  112,   60,   8;
   8:  97,  252,  184,  32;
   9: 176,  550,  540, 176, 16;
  10: 315, 1148, 1404, 640, 80;
%F Formula
The row polynomials satisfy (8*n^3-28*n^2+14*n+6+(32*n^2-108*n+76)*x+(32*n-32)*x^2)*P(n,x) = (24*n^3-92*n^2+66*n+30+(32*n^3-48*n^2-172*n+268)*x+(128*n^2-336*n+192)*x^2+(128*n-256)*x^3)*P(n-1,x) - (20*n^2-10*n-5+(72*n^2+44*n-48)*x+(64*n^2+256*n-44)*x^2+(256*n+192)*x^3+256*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[((24*n^3 - 92*n^2 + 66*n + 30 + (32*n^3 - 48*n^2 - 172*n + 268)*x + (128*n^2 - 336*n + 192)*x^2 + (128*n - 256)*x^3)*P[n - 1, x] - (20*n^2 - 10*n - 5 + (72*n^2 + 44*n - 48)*x + (64*n^2 + 256*n - 44)*x^2 + (256*n + 192)*x^3 + 256*x^4)*D[P[n - 1, x], x])/(8*n^3 - 28*n^2 + 14*n + 6 + (32*n^2 - 108*n + 76)*x + (32*n - 32)*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] = ((24*n^3 - 92*n^2 + 66*n + 30 + (32*n^3 - 48*n^2 - 172*n + 268)*x + (128*n^2 - 336*n + 192)*x^2 + (128*n - 256)*x^3)*v[i - 1] - (20*n^2 - 10*n - 5 + (72*n^2 + 44*n - 48)*x + (64*n^2 + 256*n - 44)*x^2 + (256*n + 192)*x^3 + 256*x^4)*deriv(v[i - 1], x))/(8*n^3 - 28*n^2 + 14*n + 6 + (32*n^2 - 108*n + 76)*x + (32*n - 32)*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

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.