Real-rooted atlas

A008459 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4 + 12\,x + 4\,x^{2} - 4\,n - 12\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(12\,x + 8\,x^{2} - 12\,n x - 8\,n x^{2}\right)\,P_{n-2}'(x) + \left(4\,x^{2} - 4\,n x^{2}\right)\,P_{n-2}''(x) + \left(-4 - 2\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-2\,x + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22224450
3136260968500
488627201720320165662
55944279322720085609603270628
64063628392040146419040035840268864955012
72814882870408568064038090081066240133504716812813848584
8196893428946112782166887122931227735456511795245696018432256213690102
91387554429166801210574822401272646880660895872167786304219340801466880460805123487802404
109836597229396539041411972962422020648320148188832004956483840884408448861696004477440112640102459928934012
11700701808296541650481869088918403722826400483179888870401360260915203169618636841900136963159552001312256027033620481079776927512
1250113719642995119720322459405233168618683819353666039764922563536297656320104395604684817902014873618229466112109531136037193728638976409620333379729132
13359618084323029451050264322277977432321015032273003841337808050458888823709377683232284932579328696420312473691514142720073971881984362448486410250240014909448192399056314224680
1425880599775230688053113504421118522575664164913749605696026584040230325762132760397703936951282138398592252431202017280414943787315204310111117312283222331392115324518402758246403440640163848142180162809992
Technical details
Node id
n-2da688e3fe1cbf70 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-2d57b8ff73f33801

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:        4,         2;
  2:       22,        24,          4;
  3:      136,       260,         96,          8;
  4:      886,      2720,       1720,        320,        16;
  5:     5944,     27932,      27200,       8560,       960,        32;
  6:    40636,    283920,     401464,     190400,     35840,      2688,       64;
  7:   281488,   2870408,    5680640,    3809008,   1066240,    133504,     7168,     128;
  8:  1968934,  28946112,   78216688,   71229312,  27735456,   5117952,   456960,   18432,   256;
  9: 13875544, 291668012, 1057482240, 1272646880, 660895872, 167786304, 21934080, 1466880, 46080, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = -(4*n-4+(12*n-12)*x+(4*n-4)*x^2)*P(n-2,x) - ((12*n-12)*x+(8*n-8)*x^2)*d/dx P(n-2,x) - (4*n-4)*x^2*d^2/dx^2 P(n-2,x) + (8*n-4+(4*n-2)*x)*P(n-1,x) + (4*n-2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4 + (12*n - 12)*x + (4*n - 4)*x^2)*P[n - 2, x] - ((12*n - 12)*x + (8*n - 8)*x^2)*D[P[n - 2, x], x] - (4*n - 4)*x^2*D[P[n - 2, x], {x, 2}] + (8*n - 4 + (4*n - 2)*x)*P[n - 1, x] + (4*n - 2)*x*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 4 + (12*n - 12)*x + (4*n - 4)*x^2)*v[i - 2] - ((12*n - 12)*x + (8*n - 8)*x^2)*deriv(v[i - 2], x) - (4*n - 4)*x^2*deriv(deriv(v[i - 2], x), x) + (8*n - 4 + (4*n - 2)*x)*v[i - 1] + (4*n - 2)*x*deriv(v[i - 1], x))/(n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  6. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023