Real-rooted atlas

A204420 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
2312924
315135345225720
410516809030184801102540320
59452362521735089775015961058930253628800
610395374220524947536174600126184905202951980108056025479001600
713513566216151314863551357431075775391116523963624685357036129451826146822587178291200

All 15 rows

Technical details
Node id
n-0660f7eb9cf8fcd3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f8baf88ce86e36d7

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:        1,          1;
  2:        3,         12,           9;
  3:       15,        135,         345,           225;
  4:      105,       1680,        9030,         18480,          11025;
  5:      945,      23625,      217350,        897750,        1596105,          893025;
  6:    10395,     374220,     5249475,      36174600,      126184905,       202951980,       108056025;
  7:   135135,    6621615,   131486355,    1357431075,     7753911165,     23963624685,     35703612945,      18261468225;
  8:  2027025,  129729600,  3462158700,   49945896000,   421730659350,   2104084382400,   5927369748300,    8307234936000,    4108830350625;
  9: 34459425, 2791213425, 96348552300, 1849644096300, 21603785152950, 157649595052950, 708845672234700, 1854232851170700, 2470641012464625, 1187451971330625;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(4*n^2-4*n+1)*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 - 1 + (4*n^2 - 4*n + 1)*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 - 1 + (4*n^2 - 4*n + 1)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.