Real-rooted atlas

A204420 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-2 - 6\,x + 2\,n + 12\,n x + 4\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
272180108360
3288093609720324025200
420160077616010735206350401360802822400
52177280094711680157852800126554400489888007348320457228800
63353011200160226035203056030208029907662400158968656004364902080484989120100590336000
7697426329600359423640576076063059072008604497341440563934571200021478551494404413400992003782915136028768836096000

All 15 rows

Technical details
Node id
n-a6f7ad8574b034ac · revision 0 · JSON
Definition source
derived:basis_map_A013609
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:                 4,                  6;
  2:                72,                180,                108;
  3:              2880,               9360,               9720,                3240;
  4:            201600,             776160,            1073520,              635040,              136080;
  5:          21772800,           94711680,          157852800,           126554400,            48988800,            7348320;
  6:        3353011200,        16022603520,        30560302080,         29907662400,         15896865600,         4364902080,          484989120;
  7:      697426329600,      3594236405760,      7606305907200,       8604497341440,       5639345712000,      2147855149440,       441340099200,       37829151360;
  8:   188305108992000,   1033212199219200,   2377183871462400,    3007781813836800,    2297028102969600,   1087462004428800,    312468790233600,    49934479795200,    3404623622400;
  9: 64023737057280000, 370499268851712000, 913630160617574400, 1265118571593676800, 1087783300021017600, 604033948008691200, 217160513131161600, 48849539734195200, 6250888970726400, 347271609484800;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^2+2*n-2+(12*n-6)*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[(4*n^2 + 2*n - 2 + (12*n - 6)*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] = (4*n^2 + 2*n - 2 + (12*n - 6)*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.