Real-rooted atlas

A145324 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,x + 3\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
2545112117
3648882399601989
497201711811277329436041769
51749603761643228121382312953825201044225
6367416092991249788364548534717261462892242016030282525
78817984025624641631861285221974360490795627224766903086856181440999323325

All 15 rows

Technical details
Node id
n-74d5bcdab635c3b8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5b052678045ac842

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:           6,            3;
  2:          54,           51,           12;
  3:         648,          882,          399,           60;
  4:        9720,        17118,        11277,         3294,          360;
  5:      174960,       376164,       322812,       138231,        29538,        2520;
  6:     3674160,      9299124,      9788364,      5485347,      1726146,      289224,       20160;
  7:    88179840,    256246416,    318612852,    219743604,     90795627,    22476690,     3086856,     181440;
  8:  2380855680,   7800451632,  11165011164,   9119205828,   4648917969,  1514826900,   308112012,   35767440,   1814400;
  9: 71425670400, 260202961440, 420755302872, 396391297644, 239778803178, 96582904659, 25906456260, 4462255332, 447873840, 19958400;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+3+(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[(3*n + 3 + (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] = (3*n + 3 + (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.