Real-rooted atlas

A169803 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(20 + 48\,x + 16\,x^{2} + 20\,n + 68\,n x + 40\,n x^{2} + 20\,n^{2} x + 32\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(20 + 48\,x + 16\,x^{2} + 60\,n + 304\,n x + 456\,n x^{2} + 192\,n x^{3} + 80\,n^{2} x + 240\,n^{2} x^{2} + 160\,n^{2} x^{3} + 24\,n^{3} x^{2} + 32\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 210\,x - 632\,x^{2} - 816\,x^{3} - 384\,x^{4} - 50\,n x - 280\,n x^{2} - 520\,n x^{3} - 320\,n x^{4} - 20\,n^{2} x^{2} - 72\,n^{2} x^{3} - 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
23811
35201237
48403684
5138010832233
621152264128565
734284612448801458
855520133212804003587
9899402796336016001928977
101441680568881925200115222056
112332976113041900815200518444854353
123775232220324236840880188163136132841
136109140422649152010360060480156801024324318
149871588079992192640250400177408627208192788219
Technical details
Node id
n-bbc27d1386324773 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ce70a67c11277875

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:  2,   4;
  2:  3,   8;
  3:  5,  20,   12;
  4:  8,  40,   36;
  5: 13,  80,  108,   32;
  6: 21, 152,  264,  128;
  7: 34, 284,  612,  448,   80;
  8: 55, 520, 1332, 1280,  400;
  9: 89, 940, 2796, 3360, 1600, 192;
%F Formula
The row polynomials satisfy (20*n+20+(20*n^2+68*n+48)*x+(8*n^3+32*n^2+40*n+16)*x^2)*P(n,x) = (60*n+20+(80*n^2+304*n+48)*x+(24*n^3+240*n^2+456*n+16)*x^2+(32*n^3+160*n^2+192*n)*x^3)*P(n-1,x) - (25+(50*n+210)*x+(20*n^2+280*n+632)*x^2+(72*n^2+520*n+816)*x^3+(64*n^2+320*n+384)*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[((60*n + 20 + (80*n^2 + 304*n + 48)*x + (24*n^3 + 240*n^2 + 456*n + 16)*x^2 + (32*n^3 + 160*n^2 + 192*n)*x^3)*P[n - 1, x] - (25 + (50*n + 210)*x + (20*n^2 + 280*n + 632)*x^2 + (72*n^2 + 520*n + 816)*x^3 + (64*n^2 + 320*n + 384)*x^4)*D[P[n - 1, x], x])/(20*n + 20 + (20*n^2 + 68*n + 48)*x + (8*n^3 + 32*n^2 + 40*n + 16)*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] = ((60*n + 20 + (80*n^2 + 304*n + 48)*x + (24*n^3 + 240*n^2 + 456*n + 16)*x^2 + (32*n^3 + 160*n^2 + 192*n)*x^3)*v[i - 1] - (25 + (50*n + 210)*x + (20*n^2 + 280*n + 632)*x^2 + (72*n^2 + 520*n + 816)*x^3 + (64*n^2 + 320*n + 384)*x^4)*deriv(v[i - 1], x))/(20*n + 20 + (20*n^2 + 68*n + 48)*x + (8*n^3 + 32*n^2 + 40*n + 16)*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. [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.