Real-rooted atlas

A299989 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 2\,x + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(9 + 24\,x + 22\,x^{2} + 8\,x^{3} + x^{4} + 6\,n x + 11\,n x^{2} + 6\,n x^{3} + n x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0022
10612422
20187288406224
30543246846803428482176
40162129638886000531628008641441020480
50486486018900396005058041608224807920175022012188416
601458174968359222140037017041630432508817841668550180403096312141703936
704374612363470041111320229861832863323371928253339214049705738041702683567249984201615204352

All 15 rows

Technical details
Node id
n-62408e945c925233 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d6b0c7925045f73a

Observed on the stored rows ?

  • Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 0,     2;
  1: 0,     6,     12,       4;
  2: 0,    18,     72,      88,       40,        6;
  3: 0,    54,    324,     684,      680,      342,        84,         8;
  4: 0,   162,   1296,    3888,     6000,     5316,      2800,       864,       144,        10;
  5: 0,   486,   4860,   18900,    39600,    50580,     41608,     22480,      7920,      1750,       220,       12;
  6: 0,  1458,  17496,   83592,   221400,   370170,    416304,    325088,    178416,     68550,     18040,     3096,      312,       14;
  7: 0,  4374,  61236,  347004,  1111320,  2298618,   3286332,   3371928,   2533392,   1404970,    573804,   170268,    35672,     4998,     420,     16;
  8: 0, 13122, 209952, 1376352,  5171040, 12750696,  22141728,  28203840,  26999136,  19689340,  10999648,  4700640,  1522976,   367304,   63840,   7552,    544,    18;
  9: 0, 39366, 708588, 5275044, 22744800, 65137608, 133167888, 202831776, 236382624, 214318980, 152630984, 85727592, 37937952, 13146504, 3522960, 714816, 106080, 10854, 684, 20;
%F Formula
The row polynomials satisfy (3+(2*n+2)*x+n*x^2)*P(n,x) = (9+(6*n+24)*x+(11*n+22)*x^2+(6*n+8)*x^3+(n+1)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 2*x.
%t Mathematica
P[0, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((9 + (6*n + 24)*x + (11*n + 22)*x^2 + (6*n + 8)*x^3 + (n + 1)*x^4)*P[n - 1, x])/(3 + (2*n + 2)*x + n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((9 + (6*n + 24)*x + (11*n + 22)*x^2 + (6*n + 8)*x^3 + (n + 1)*x^4)*v[i - 1])/(3 + (2*n + 2)*x + n*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.