Real-rooted atlas

basis-map coefficients of A297191, shifted → θ+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(1 + 3\,x + x^{2} + 2\,n + 6\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(3\,x + 2\,x^{2} + 6\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(x^{2} + 2\,n x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
139315
2575125505260
374412107245098014776139
4922952722569930625592381441583249190323
5111125331351116032502714030197435770385713068012705605117464270
61353235340112532922890976105131211770567430051224694956467352651122531603101413359377681
71524574535626515635339250316150380063103990956131119995324941760010069273501899397502219217015970506825015751520744378175

All 15 rows

Technical details
Node id
n-03edc5c28196c3db · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-6a8026fb20183be6

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:  3,       9,          3;
  2:  5,      75,        125,           50,             5;
  3:  7,     441,       2107,         2450,           980,            147,              7;
  4:  9,    2295,      27225,        69930,         62559,          23814,           4158,            324,              9;
  5: 11,   11253,     313511,      1603250,       2714030,        1974357,         703857,         130680,          12705,           605,            11;
  6: 13,   53235,    3401125,     32922890,      97610513,      121177056,       74300512,       24694956,        4673526,        511225,         31603,         1014,          13;
  7: 15,  245745,   35626515,    635339250,    3161503800,     6310399095,     6131119995,     3249417600,     1006927350,     189939750,      22192170,      1597050,       68250,       1575,        15;
  8: 17, 1114095,  364783025,  11803733930,   96096778067,   297597748266,   437037781362,   347071915476,   161989979151,   46891680550,    8705036626,   1054379508,    83473026,    4248300,    132940,     2312,     17;
  9: 19, 4980717, 3675513919, 213933672050, 2804196409490, 13168550433801, 28365568061941, 32485269066120, 21747677909295, 9068643641915, 2458851138029, 445872074700, 54956344040, 4627973070, 264418782, 10015584, 239343, 3249, 19;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(6*n+3)*x+(2*n+1)*x^2)*P(n-1,x) + ((6*n+3)*x+(4*n+2)*x^2)*d/dx P(n-1,x) + (2*n+1)*x^2*d^2/dx^2 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[((2*n + 1 + (6*n + 3)*x + (2*n + 1)*x^2)*P[n - 1, x] + ((6*n + 3)*x + (4*n + 2)*x^2)*D[P[n - 1, x], x] + (2*n + 1)*x^2*D[P[n - 1, x], {x, 2}])/(2*n - 1)]];
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] = ((2*n + 1 + (6*n + 3)*x + (2*n + 1)*x^2)*v[i - 1] + ((6*n + 3)*x + (4*n + 2)*x^2)*deriv(v[i - 1], x) + (2*n + 1)*x^2*deriv(deriv(v[i - 1], x), x))/(2*n - 1)); 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

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. [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.