Real-rooted atlas

A136572 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(n + 6\,n x + 9\,n x^{2} + 3\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
214481880
3603786481621248
43122880907210368194424576
519202340010800022680019440029160583680
613680207360126360038880006123600419904052488016220160
71108802010960152409606191640014288400018003384010287648011022480516096000

All 15 rows

Technical details
Node id
n-c4eea4ee44016286 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1f45c55ca90e4ac8

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,         3;
  2:       14,        48,         18;
  3:       60,       378,        648,         162;
  4:      312,      2880,       9072,       10368,        1944;
  5:     1920,     23400,     108000,      226800,      194400,        29160;
  6:    13680,    207360,    1263600,     3888000,     6123600,      4199040,       524880;
  7:   110880,   2010960,   15240960,    61916400,   142884000,    180033840,    102876480,     11022480;
  8:  1008000,  21288960,  193052160,   975421440,  2971987200,   5486745600,   5761082880,   2821754880,   264539520;
  9: 10160640, 244944000, 2586608640, 15637224960, 59256852480, 144438577920, 222213196800, 199991877120, 85710804480, 7142567040;
%F Formula
The row polynomials satisfy (3*n-2+3*x)*P(n,x) = (3*n^2+n+(9*n^2+6*n)*x+9*n*x^2)*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[Together[((3*n^2 + n + (9*n^2 + 6*n)*x + 9*n*x^2)*P[n - 1, x])/(3*n - 2 + 3*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 + n + (9*n^2 + 6*n)*x + 9*n*x^2)*v[i - 1])/(3*n - 2 + 3*x)); 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.