Real-rooted atlas

A088996 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-1 + 3\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
2109221
380102436231
48801442881238243465
51232024588195447737152612065835
620944049191647977624879372364111967201514205
7418880011304400130389328334292318883173046892772504040883535

All 15 rows

Technical details
Node id
n-fb1d0af449b91ab1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a80b2260535f18e7

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:          2,          1;
  2:         10,          9,           2;
  3:         80,        102,          43,           6;
  4:        880,       1442,         881,         238,         24;
  5:      12320,      24588,       19544,        7737,       1526,        120;
  6:     209440,     491916,      479776,      248793,      72364,      11196,       720;
  7:    4188800,   11304400,    13038932,     8334292,    3188831,     730468,     92772,     5040;
  8:   96342400,  293511600,   390330636,   296000172,  140017449,   42311412,   7977500,   858096,   40320;
  9: 2504902400, 8498383200, 12790200936, 11208980196, 6304455222, 2360253753, 588217708, 94107996, 8771184, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-1+n*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[(3*n - 1 + n*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] = (3*n - 1 + n*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.