Real-rooted atlas

A120910 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(9 + 12\,x + 4\,x^{2} + 6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
115621
2636012135
324337818024825
489119441512480484875
5315989109720504012009628125
610935379085346038880151202880192159375
737179153090265356249480136080423366720384890625

All 15 rows

Technical details
Node id
n-37227a36b7a07022 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7ad822d7676abaf6

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:      3;
  1:     15,       6;
  2:     63,      60,      12;
  3:    243,     378,     180,      24;
  4:    891,    1944,    1512,     480,      48;
  5:   3159,    8910,    9720,    5040,    1200,      96;
  6:  10935,   37908,   53460,   38880,   15120,    2880,     192;
  7:  37179,  153090,  265356,  249480,  136080,   42336,    6720,    384;
  8: 124659,  594864, 1224720, 1415232,  997920,  435456,  112896,  15360,   768;
  9: 413343, 2243862, 5353776, 7348320, 6368544, 3592512, 1306368, 290304, 34560, 1536;
%F Formula
The row polynomials satisfy (2*n+1+2*x)*P(n,x) = (6*n+9+(4*n+12)*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 9 + (4*n + 12)*x + 4*x^2)*P[n - 1, x])/(2*n + 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 9 + (4*n + 12)*x + 4*x^2)*v[i - 1])/(2*n + 1 + 2*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.