Real-rooted atlas

A099040 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n + 16\,n x + 4\,n x^{2} + 6\,n^{2} x + 5\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(48 + 100\,x + 68\,x^{2} + 16\,x^{3} - 24\,n + 10\,n x + 48\,n x^{2} + 20\,n x^{3} - 18\,n^{2} x - 6\,n^{2} x^{2} + 4\,n^{2} x^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(36 + 108\,x + 116\,x^{2} + 52\,x^{3} + 8\,x^{4} + 36\,n x + 78\,n x^{2} + 52\,n x^{3} + 10\,n x^{4} + 6\,n^{2} x^{2} + 8\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26201238
316809632224
444288516352801280
51209762328240011201927136
632831849480130889280326444839072
7896101123609662464595203187289601024210944
82448314881310402726403264802353921008002355223041126144
96688965764590721115904160976014609288431363000325990451205957120
10182722926721564416435072073438408039616584595228047368524801484801126431272448
1149920878336521472016329216315737604048972835546112214988808810496233472036044824576163110912
12136384261478417071296594375681295350401904609281960192641432176647400908826439680620646486016053248846061568
133726087731456550513922109629445116345608485624321002015616858725376536251392241592320764375041609728020234241146884367572992
1410179842272844817528409673312819219583404803616853760482045760047422515203474892800189219840075495833621422899240894464470220824576022452183040
Technical details
Node id
n-8fac5e5e83554770 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-831f2354c620f023

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:    2,     4;
  2:    6,    20,     12;
  3:   16,    80,     96,      32;
  4:   44,   288,    516,     352,      80;
  5:  120,   976,   2328,    2400,    1120,     192;
  6:  328,  3184,   9480,   13088,    9280,    3264,    448;
  7:  896, 10112,  36096,   62464,   59520,   31872,   8960,   1024;
  8: 2448, 31488, 131040,  272640,  326480,  235392, 100800,  23552,  2304;
  9: 6688, 96576, 459072, 1115904, 1609760, 1460928, 843136, 300032, 59904, 5120;
%F Formula
The row polynomials satisfy (12*n+(6*n^2+16*n)*x+(n^3+5*n^2+4*n)*x^2)*P(n,x) = -(24*n-48+(18*n^2-10*n-100)*x+(2*n^3+6*n^2-48*n-68)*x^2-(4*n^2+20*n+16)*x^3)*P(n-1,x) + (36+(36*n+108)*x+(6*n^2+78*n+116)*x^2+(8*n^2+52*n+52)*x^3+(2*n^2+10*n+8)*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 - 48 + (18*n^2 - 10*n - 100)*x + (2*n^3 + 6*n^2 - 48*n - 68)*x^2 - (4*n^2 + 20*n + 16)*x^3)*P[n - 1, x] + (36 + (36*n + 108)*x + (6*n^2 + 78*n + 116)*x^2 + (8*n^2 + 52*n + 52)*x^3 + (2*n^2 + 10*n + 8)*x^4)*D[P[n - 1, x], x])/(12*n + (6*n^2 + 16*n)*x + (n^3 + 5*n^2 + 4*n)*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 - 48 + (18*n^2 - 10*n - 100)*x + (2*n^3 + 6*n^2 - 48*n - 68)*x^2 - (4*n^2 + 20*n + 16)*x^3)*v[i - 1] + (36 + (36*n + 108)*x + (6*n^2 + 78*n + 116)*x^2 + (8*n^2 + 52*n + 52)*x^3 + (2*n^2 + 10*n + 8)*x^4)*deriv(v[i - 1], x))/(12*n + (6*n^2 + 16*n)*x + (n^3 + 5*n^2 + 4*n)*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.