Real-rooted atlas

A178821 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n + n x + n^{2}\right)\,P_{n}(x) = \left(16 + 16\,x + 4\,x^{2} + 12\,n + 8\,n x + n x^{2} + 2\,n^{2} + n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
115520
21209015225
3700840315351890
43360560033608407013230
514112302402520010080189012681648
65376014112015120084000252003780210459270
71900805913607761605544002310005544069303302405700

All 15 rows

Technical details
Node id
n-1cc2169a82f1af75 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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:      15,       5;
  2:     120,      90,       15;
  3:     700,     840,      315,       35;
  4:    3360,    5600,     3360,      840,       70;
  5:   14112,   30240,    25200,    10080,     1890,     126;
  6:   53760,  141120,   151200,    84000,    25200,    3780,     210;
  7:  190080,  591360,   776160,   554400,   231000,   55440,    6930,    330;
  8:  633600, 2280960,  3548160,  3104640,  1663200,  554400,  110880,  11880,   495;
  9: 2013440, 8236800, 14826240, 15375360, 10090080, 4324320, 1201200, 205920, 19305, 715;
%F Formula
The row polynomials satisfy (n^2+n+n*x)*P(n,x) = (2*n^2+12*n+16+(n^2+8*n+16)*x+(n+4)*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[((2*n^2 + 12*n + 16 + (n^2 + 8*n + 16)*x + (n + 4)*x^2)*P[n - 1, x])/(n^2 + n + n*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] = ((2*n^2 + 12*n + 16 + (n^2 + 8*n + 16)*x + (n + 4)*x^2)*v[i - 1])/(n^2 + n + n*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.