Real-rooted atlas

A225470 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 9\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16915
25413581270
3648210621877295670
497203742251759306186561136080
517496076107612684601016955393660590493674160
63674160175572363348734432772195174194554782969531441110224800
78817984045444110496171138010879187767130166752715663515580130547829693637418400

All 15 rows

Technical details
Node id
n-f6ce73d04c355dca · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dd8e883ba3f13bd7

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,            9;
  2:          54,          135,            81;
  3:         648,         2106,          2187,           729;
  4:        9720,        37422,         51759,         30618,          6561;
  5:      174960,       761076,       1268460,       1016955,        393660,        59049;
  6:     3674160,     17557236,      33487344,      32772195,      17419455,      4782969,       531441;
  7:    88179840,    454441104,     961711380,    1087918776,     713016675,    271566351,     55801305,     4782969;
  8:  2380855680,  13063528368,   30056177196,   38029209372,   29042719209,  13749441552,   3950732394,   631351908,   43046721;
  9: 71425670400, 413333552160, 1019257071192, 1411381875924, 1213544460618, 673867719441, 242266945788, 54497148786, 6973568802, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+3+9*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 + 9*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 + 9*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.