Real-rooted atlas

A145324 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
224341270
319239226660910
4192050725012219636014560
523040743049564861436196922520276640
6322560122457619335041625288767176192816201606086080
751609602249625641957248434061442690240899896402057904181440152152000

All 15 rows

Technical details
Node id
n-b375e1ad47995c78 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-09a35ffebf310436

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:          4,           3;
  2:         24,          34,          12;
  3:        192,         392,         266,          60;
  4:       1920,        5072,        5012,        2196,         360;
  5:      23040,       74304,       95648,       61436,       19692,        2520;
  6:     322560,     1224576,     1933504,     1625288,      767176,      192816,      20160;
  7:    5160960,    22496256,    41957248,    43406144,    26902408,     9989640,    2057904,     181440;
  8:   92897280,   456542208,   980193024,  1200883072,   918304784,   448837600,  136938672,   23844960,   1814400;
  9: 1857945600, 10152714240, 24625824768, 34799784704, 31575809472, 19078104624, 7675987040, 1983224592, 298582560, 19958400;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+2+(n+2)*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[(2*n + 2 + (n + 2)*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] = (2*n + 2 + (n + 2)*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

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