Real-rooted atlas

A008277 → θ+1 → D+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 3\,x + x^{3}\right)\,P_{n-3}(x) + \left(-2\,x - 3\,x^{2}\right)\,P_{n-3}'(x) + \left(-x^{2} - x^{3}\right)\,P_{n-3}''(x) + \left(-11 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(2\,x\right)\,P_{n-2}'(x) + \left(x^{2}\right)\,P_{n-2}''(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
\(P_{3}(x) = 12 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
222
312618
4506012122
518039018020770
660221001680420304832
71932102061260053208404230940
8605046620834125292013860151256204430
918660204630510300456540176400315002520721400622
1057002874500296076035897401919610498960646803960909969302
11173052366900616552800264730401882188066791341247400122760594011073745122
125232501519518090102012186426240171462720797837042013211228314002187908580132566684120
1315779406235047048090042012687274601479878400878317440287567280541981445945940370370120121564519846032
144750202254135700252920304084138654601226223999090983692803759195440911710800133153020117117006006001638018237378951794
1514283372103079340613154286600546985311609845647812089954090226459639180001395035928026043217203032829802186184093912021840210320153167874
Technical details
Node id
n-87306d44180b621e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2f80f29c34e4f8b9

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:     0;
   2:     2;
   3:    12,      6;
   4:    50,     60,      12;
   5:   180,    390,     180,      20;
   6:   602,   2100,    1680,     420,      30;
   7:  1932,  10206,   12600,    5320,     840,     42;
   8:  6050,  46620,   83412,   52920,   13860,   1512,    56;
   9: 18660, 204630,  510300,  456540,  176400,  31500,  2520,   72;
  10: 57002, 874500, 2960760, 3589740, 1919610, 498960, 64680, 3960, 90;
%F Formula
The row polynomials satisfy P(n,x) = (6+3*x+x^3)*P(n-3,x) - (2*x+3*x^2)*d/dx P(n-3,x) - (x^2+x^3)*d^2/dx^2 P(n-3,x) - (11+6*x+3*x^2)*P(n-2,x) + 2*x*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 2, P(3,x) = 12 + 6*x.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 12 + 6*x;
P[n_, x_] := P[n, x] = Expand[(6 + 3*x + x^3)*P[n - 3, x] - (2*x + 3*x^2)*D[P[n - 3, x], x] - (x^2 + x^3)*D[P[n - 3, x], {x, 2}] - (11 + 6*x + 3*x^2)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; v[3] = 12 + 6*x; for(n = 4, N, my(i = n); v[i] = (6 + 3*x + x^3)*v[i - 3] - (2*x + 3*x^2)*deriv(v[i - 3], x) - (x^2 + x^3)*deriv(deriv(v[i - 3], x), x) - (11 + 6*x + 3*x^2)*v[i - 2] + 2*x*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), x) + (6 + 3*x)*v[i - 1]); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.