Real-rooted atlas

A169803 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-14 - 12\,x + 32\,x^{2} - 2\,n + 20\,n x + 32\,n x^{2} + 20\,n^{2} + 32\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(-14 - 12\,x + 32\,x^{2} - 14\,n + 20\,n x + 176\,n x^{2} + 128\,n x^{3} + 52\,n^{2} + 144\,n^{2} x + 128\,n^{2} x^{2} + 24\,n^{3} + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(-55 - 328\,x - 724\,x^{2} - 704\,x^{3} - 256\,x^{4} - 70\,n - 332\,n x - 512\,n x^{2} - 256\,n x^{3} - 20\,n^{2} - 72\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1426
27411
31518437
428441284
553112608233
69725218432565
7176550540176161458

All 15 rows

Technical details
Node id
n-e3d942ce710a27f2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-594a37631f56a83b

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   4,    2;
  2:   7,    4;
  3:  15,   18,    4;
  4:  28,   44,   12;
  5:  53,  112,   60,    8;
  6:  97,  252,  184,   32;
  7: 176,  550,  540,  176,  16;
  8: 315, 1148, 1404,  640,  80;
  9: 559, 2334, 3452, 2120, 480, 32;
%F Formula
The row polynomials satisfy (8*n^3+20*n^2-2*n-14+(32*n^2+20*n-12)*x+(32*n+32)*x^2)*P(n,x) = (24*n^3+52*n^2-14*n-14+(32*n^3+144*n^2+20*n-12)*x+(128*n^2+176*n+32)*x^2+128*n*x^3)*P(n-1,x) - (20*n^2+70*n+55+(72*n^2+332*n+328)*x+(64*n^2+512*n+724)*x^2+(256*n+704)*x^3+256*x^4)*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[((24*n^3 + 52*n^2 - 14*n - 14 + (32*n^3 + 144*n^2 + 20*n - 12)*x + (128*n^2 + 176*n + 32)*x^2 + 128*n*x^3)*P[n - 1, x] - (20*n^2 + 70*n + 55 + (72*n^2 + 332*n + 328)*x + (64*n^2 + 512*n + 724)*x^2 + (256*n + 704)*x^3 + 256*x^4)*D[P[n - 1, x], x])/(8*n^3 + 20*n^2 - 2*n - 14 + (32*n^2 + 20*n - 12)*x + (32*n + 32)*x^2)]];
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] = ((24*n^3 + 52*n^2 - 14*n - 14 + (32*n^3 + 144*n^2 + 20*n - 12)*x + (128*n^2 + 176*n + 32)*x^2 + 128*n*x^3)*v[i - 1] - (20*n^2 + 70*n + 55 + (72*n^2 + 332*n + 328)*x + (64*n^2 + 512*n + 724)*x^2 + (256*n + 704)*x^3 + 256*x^4)*deriv(v[i - 1], x))/(8*n^3 + 20*n^2 - 2*n - 14 + (32*n^2 + 20*n - 12)*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 - 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.