Real-rooted atlas

A156992 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
29612024240
31536364817281927104
430720124800105600268801920289920
57372804861440656640028800004608002304015528960
6206438402145024004357785602935296007902720087091203225601052513280
7660602880106264166403132186624030454824960122830848002239856640180633600516096087772446720

All 15 rows

Technical details
Node id
n-0b75221cfe4c7721 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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:            8,              4;
  2:           96,            120,              24;
  3:         1536,           3648,            1728,             192;
  4:        30720,         124800,          105600,           26880,            1920;
  5:       737280,        4861440,         6566400,         2880000,          460800,           23040;
  6:     20643840,      214502400,       435778560,       293529600,        79027200,         8709120,         322560;
  7:    660602880,    10626416640,     31321866240,     30454824960,     12283084800,      2239856640,      180633600,       5160960;
  8:  23781703680,   585717350400,   2446449868800,   3304727838720,   1874760007680,    503317463040,    66328657920,    4087480320,     92897280;
  9: 951268147200, 35618675097600, 207430336512000, 379401781248000, 291065757696000, 107959644979200, 20678934528000, 2062319616000, 100329062400, 1857945600;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+4+(2*n+2)*x)*P(n-1,x) + (2*n+2)*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[(4*n + 4 + (2*n + 2)*x)*P[n - 1, x] + (2*n + 2)*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] = (4*n + 4 + (2*n + 2)*x)*v[i - 1] + (2*n + 2)*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.