Real-rooted atlas

A094638 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 5\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
161218
266276288630
310566792145441036832760
4221761933206314409158404976642260440
5576576635688028016640616982406788966429859840194397840
6178738562385767521326211200392984352065468528645813710848214990848020022977520
764345881610090166976677840503682528761075205657935587847592292311045657484165121805923123202402757302400

All 15 rows

Technical details
Node id
n-3941bc124b996696 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-5095f420bdb318b4

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,             12;
  2:            66,            276,             288;
  3:          1056,           6792,           14544,            10368;
  4:         22176,         193320,          631440,           915840,           497664;
  5:        576576,        6356880,        28016640,         61698240,         67889664,         29859840;
  6:      17873856,      238576752,      1326211200,       3929843520,       6546852864,       5813710848,        2149908480;
  7:     643458816,    10090166976,     67784050368,     252876107520,     565793558784,     759229231104,      565748416512,      180592312320;
  8:   26381811456,   475468892352,   3747802094784,   16875189243648,   47473642232064,   85444580118528,    96081691262976,    61716132790272,   17336861982720;
  9: 1213563326976, 24720804685440, 223749536734080, 1181021331444480, 4006307980988928, 9057604046515200, 13647772450897920, 13215764764753920, 7462837992554496, 1872381094133760;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+1+12*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[(5*n + 1 + 12*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] = (5*n + 1 + 12*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

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.