Real-rooted atlas

A063967 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-12 - 10\,x + 12\,n + 4\,n x - 3\,n x^{2} + 6\,n^{2} x + 2\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(72 + 72\,x + 16\,x^{2} - 24\,n + 46\,n x + 54\,n x^{2} + 12\,n x^{3} - 18\,n^{2} x + 4\,n^{2} x^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(36 + 72\,x + 44\,x^{2} + 8\,x^{3} + 36\,n x + 66\,n x^{2} + 36\,n x^{3} + 6\,n x^{4} + 6\,n^{2} x^{2} + 8\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
36201238
416809632224
544288516352801280
61209762328240011201927136
732831849480130889280326444839072
8896101123609662464595203187289601024210944
92448314881310402726403264802353921008002355223041126144
106688965764590721115904160976014609288431363000325990451205957120
11182722926721564416435072073438408039616584595228047368524801484801126431272448
1249920878336521472016329216315737604048972835546112214988808810496233472036044824576163110912
13136384261478417071296594375681295350401904609281960192641432176647400908826439680620646486016053248846061568
143726087731456550513922109629445116345608485624321002015616858725376536251392241592320764375041609728020234241146884367572992
1510179842272844817528409673312819219583404803616853760482045760047422515203474892800189219840075495833621422899240894464470220824576022452183040
Technical details
Node id
n-9be176da304a3877 · 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 -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:    2,     4;
   3:    6,    20,     12;
   4:   16,    80,     96,      32;
   5:   44,   288,    516,     352,      80;
   6:  120,   976,   2328,    2400,    1120,     192;
   7:  328,  3184,   9480,   13088,    9280,    3264,    448;
   8:  896, 10112,  36096,   62464,   59520,   31872,   8960,   1024;
   9: 2448, 31488, 131040,  272640,  326480,  235392, 100800,  23552,  2304;
  10: 6688, 96576, 459072, 1115904, 1609760, 1460928, 843136, 300032, 59904, 5120;
%F Formula
The row polynomials satisfy (12*n-12+(6*n^2+4*n-10)*x+(n^3+2*n^2-3*n)*x^2)*P(n,x) = -(24*n-72+(18*n^2-46*n-72)*x+(2*n^3-54*n-16)*x^2-(4*n^2+12*n)*x^3)*P(n-1,x) + (36+(36*n+72)*x+(6*n^2+66*n+44)*x^2+(8*n^2+36*n+8)*x^3+(2*n^2+6*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[(-(24*n - 72 + (18*n^2 - 46*n - 72)*x + (2*n^3 - 54*n - 16)*x^2 - (4*n^2 + 12*n)*x^3)*P[n - 1, x] + (36 + (36*n + 72)*x + (6*n^2 + 66*n + 44)*x^2 + (8*n^2 + 36*n + 8)*x^3 + (2*n^2 + 6*n)*x^4)*D[P[n - 1, x], x])/(12*n - 12 + (6*n^2 + 4*n - 10)*x + (n^3 + 2*n^2 - 3*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] = (-(24*n - 72 + (18*n^2 - 46*n - 72)*x + (2*n^3 - 54*n - 16)*x^2 - (4*n^2 + 12*n)*x^3)*v[i - 1] + (36 + (36*n + 72)*x + (6*n^2 + 66*n + 44)*x^2 + (8*n^2 + 36*n + 8)*x^3 + (2*n^2 + 6*n)*x^4)*deriv(v[i - 1], x))/(12*n - 12 + (6*n^2 + 4*n - 10)*x + (n^3 + 2*n^2 - 3*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.