Real-rooted atlas

A052553 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-20 - 8\,x + 20\,n - 12\,n x + 8\,n x^{2} + 20\,n^{2} x - 16\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-100 - 240\,x - 128\,x^{2} + 60\,n - 16\,n x - 216\,n x^{2} - 64\,n x^{3} + 80\,n^{2} x + 96\,n^{2} x^{2} - 32\,n^{2} x^{3} + 24\,n^{3} x^{2} + 32\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 110\,x - 152\,x^{2} - 64\,x^{3} - 50\,n x - 200\,n x^{2} - 232\,n x^{3} - 64\,n x^{4} - 20\,n^{2} x^{2} - 72\,n^{2} x^{3} - 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3246
43811
55201237
68403684
7138010832233
821152264128565
934284612448801458
1055520133212804003587
11899402796336016001928977
121441680568881925200115222056
132332976113041900815200518444854353
143775232220324236840880188163136132841
156109140422649152010360060480156801024324318
Technical details
Node id
n-92952eac23160a9d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ce70a67c11277875

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:  2,   4;
   4:  3,   8;
   5:  5,  20,   12;
   6:  8,  40,   36;
   7: 13,  80,  108,   32;
   8: 21, 152,  264,  128;
   9: 34, 284,  612,  448,  80;
  10: 55, 520, 1332, 1280, 400;
%F Formula
The row polynomials satisfy (20*n-20+(20*n^2-12*n-8)*x+(8*n^3-16*n^2+8*n)*x^2)*P(n,x) = (60*n-100+(80*n^2-16*n-240)*x+(24*n^3+96*n^2-216*n-128)*x^2+(32*n^3-32*n^2-64*n)*x^3)*P(n-1,x) - (25+(50*n+110)*x+(20*n^2+200*n+152)*x^2+(72*n^2+232*n+64)*x^3+(64*n^2+64*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[((60*n - 100 + (80*n^2 - 16*n - 240)*x + (24*n^3 + 96*n^2 - 216*n - 128)*x^2 + (32*n^3 - 32*n^2 - 64*n)*x^3)*P[n - 1, x] - (25 + (50*n + 110)*x + (20*n^2 + 200*n + 152)*x^2 + (72*n^2 + 232*n + 64)*x^3 + (64*n^2 + 64*n)*x^4)*D[P[n - 1, x], x])/(20*n - 20 + (20*n^2 - 12*n - 8)*x + (8*n^3 - 16*n^2 + 8*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] = ((60*n - 100 + (80*n^2 - 16*n - 240)*x + (24*n^3 + 96*n^2 - 216*n - 128)*x^2 + (32*n^3 - 32*n^2 - 64*n)*x^3)*v[i - 1] - (25 + (50*n + 110)*x + (20*n^2 + 200*n + 152)*x^2 + (72*n^2 + 232*n + 64)*x^3 + (64*n^2 + 64*n)*x^4)*deriv(v[i - 1], x))/(20*n - 20 + (20*n^2 - 12*n - 8)*x + (8*n^3 - 16*n^2 + 8*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

Same fingerprint ?

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.