Real-rooted atlas

A145324 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(6 + 4\,x + 5\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111819
217626096532
3369682766176153619684
49609628909632609616345630720905464
52978976112682801704728012893440487526473728049800520
6107243136489069408929213920941487680536525824163049472206438403187233280
7439696857623483626080537479917766833584032052125164544238538547206063980544660602880232668029440

All 15 rows

Technical details
Node id
n-4c1ffa8a6e81d9b0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1a6646c65534e500

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              1;
  1:             11,              8;
  2:            176,            260,              96;
  3:           3696,           8276,            6176,            1536;
  4:          96096,         289096,          326096,          163456,           30720;
  5:        2978976,       11268280,        17047280,        12893440,         4875264,          737280;
  6:      107243136,      489069408,       929213920,       941487680,       536525824,       163049472,        20643840;
  7:     4396968576,    23483626080,     53747991776,     68335840320,     52125164544,     23853854720,      6063980544,      660602880;
  8:   202260554496,  1238537668416,   3317818160576,   5078376358656,   4857847820544,   2973783240704,   1137681874944,   248691032064,    23781703680;
  9: 10315288279296, 71255843269056, 218750232926016, 391709920714496, 450885293193984, 345976858097664, 176973105250304, 58190517633024, 11160508170240, 951268147200;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+6+(4*n+4)*x)*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 + 6 + (4*n + 4)*x)*P[n - 1, 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 + 6 + (4*n + 4)*x)*v[i - 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

Proof chains ? JSON

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.