Real-rooted atlas

A063967 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
32022446
496192808376
546415161100248163344
62240113681285649207043232120
710816827121374167959218976188864331464
852224590592139033611456644051846624048641283655232

All 15 rows

Technical details
Node id
n-79ed04620ce13b77 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b9ca97c881366b5c

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 -1)
  • 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:       1;
   2:       4,        2;
   3:      20,       22,         4;
   4:      96,      192,        80,         8;
   5:     464,     1516,      1100,       248,        16;
   6:    2240,    11368,     12856,      4920,       704,       32;
   7:   10816,    82712,    137416,     79592,     18976,     1888,      64;
   8:   52224,   590592,   1390336,   1145664,    405184,    66240,    4864,    128;
   9:  252160,  4165904,  13568624,  15308064,   7540432,  1802112,  215104,  12160,   256;
  10: 1217536, 29147744, 129184672, 194455200, 128254848, 42048160, 7256192, 661120, 29696, 512;
%F Formula
The row polynomials satisfy P(n,x) = (4+2*x)*P(n-2,x) + 2*x*d/dx P(n-2,x) + (4+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[(4 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + (4 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = (4 + 2*x)*v[i - 2] + 2*x*deriv(v[i - 2], x) + (4 + 2*x)*v[i - 1] + 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, v[i])))};
T(10)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [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