Real-rooted atlas

A196347 → x ↦ x+1 → basis A013610

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
218361872
31624864861621296
419447776116647776194431104
52916014580029160029160014580029160933120
652488031492807873200104976007873200314928052488033592320
7110224807715736023147208038578680038578680023147208077157360110224801410877440

All 15 rows

Technical details
Node id
n-df0f1b594ece8aed · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:          3,           3;
  2:         18,          36,           18;
  3:        162,         486,          486,          162;
  4:       1944,        7776,        11664,         7776,         1944;
  5:      29160,      145800,       291600,       291600,       145800,        29160;
  6:     524880,     3149280,      7873200,     10497600,      7873200,      3149280,       524880;
  7:   11022480,    77157360,    231472080,    385786800,    385786800,    231472080,     77157360,     11022480;
  8:  264539520,  2116316160,   7407106560,  14814213120,  18517766400,  14814213120,   7407106560,   2116316160,   264539520;
  9: 7142567040, 64283103360, 257132413440, 599975631360, 899963447040, 899963447040, 599975631360, 257132413440, 64283103360, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+3*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[(3*n + 3*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] = (3*n + 3*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

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.