Real-rooted atlas

A008955 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
2525816126
31456256014922885796
471344172032155028618569216469476
55422144166416322038372812452456379321646080059153976
659101369622043322563420023856282494612013100373763233387523317760010706869656
7874700270083841605160967221880917767532543636484707302514081762377981443659748249632514048002633889935376

All 15 rows

Technical details
Node id
n-848ee0626a262d9f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-95f9dd9edf6a7056

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,                 2;
  2:               52,                58,                16;
  3:             1456,              2560,              1492,               288;
  4:            71344,            172032,            155028,             61856,              9216;
  5:          5422144,          16641632,          20383728,          12452456,           3793216,            460800;
  6:        591013696,        2204332256,        3420023856,        2824946120,        1310037376,         323338752,          33177600;
  7:      87470027008,      384160516096,      722188091776,      753254363648,      470730251408,      176237798144,       36597482496,       3251404800;
  8:   16881715212544,    85339143063552,   188554847773056,   237818167931392,   187267497068688,    94267367222016,    29621752284160,    5311998885888,    416179814400;
  9: 4119138511860736, 23557588771938816, 59832324032921088, 88573518314494720, 84219812489645376, 53338572127299360, 22499021047301632, 6094851598190592, 962091694227456, 67421129932800;
%F Formula
The row polynomials satisfy P(n,x) = (3*n^2+1+2*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^2 + 1 + 2*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^2 + 1 + 2*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

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.