Real-rooted atlas

A136572 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5\,n + 4\,n x\right)\,P_{n-1}(x) + \left(4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
25011232194
3750362425923847350
415000142464192768675846144423960
53750006710880153792009600000199680012288034184760
611250000371387520136063872013483008004819968006635520029491203642878160
73937500002371241376013421697408019854567936010860595200025108070400245661696082575360493122031920

All 15 rows

Technical details
Node id
n-2ccf2fd4c3930c32 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c7f57c7d3a795bc5

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:            5,               4;
  2:           50,             112,               32;
  3:          750,            3624,             2592,              384;
  4:        15000,          142464,           192768,            67584,             6144;
  5:       375000,         6710880,         15379200,          9600000,          1996800,           122880;
  6:     11250000,       371387520,       1360638720,       1348300800,        481996800,         66355200,         2949120;
  7:    393750000,     23712413760,     134216974080,     198545679360,     108605952000,      25108070400,      2456616960,       82575360;
  8:  15750000000,   1719893790720,   14717362544640,   31297155563520,   24599261675520,    8497004544000,   1373393387520,   100411637760,    2642411520;
  9: 708750000000, 139878397048320, 1783847594188800, 5318289852825600, 5775958056960000, 2797399442718720, 664347837726720, 79264418365440, 4494741995520, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+4*n*x)*P(n-1,x) + 4*n*x*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[(5*n + 4*n*x)*P[n - 1, x] + 4*n*x*D[P[n - 1, x], x]];
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] = (5*n + 4*n*x)*v[i - 1] + 4*n*x*deriv(v[i - 1], x)); 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. [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.