Real-rooted atlas

A204420 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-48 + 24\,x - 3\,x^{2} + 176\,n - 76\,n x + 8\,n x^{2} - 204\,n^{2} + 64\,n^{2} x - 4\,n^{2} x^{2} + 96\,n^{3} - 16\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(6 - 2\,x - 16\,n + 4\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
2612321
312030013515570
45040142808400168010529505
53628801118880774900207900236259452489130
63991680013106016010170468032848200509355037422010395311008005
76227020800214832217601816214400066897230401267566300128648520662161513513553965081170

All 15 rows

Technical details
Node id
n-f618e014540b976e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6ffafa6a7d4d1aed

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1;
  1:               1,                1;
  2:               6,               12,                3;
  3:             120,              300,              135,              15;
  4:            5040,            14280,             8400,            1680,             105;
  5:          362880,          1118880,           774900,          207900,           23625,            945;
  6:        39916800,        131060160,        101704680,        32848200,         5093550,         374220,         10395;
  7:      6227020800,      21483221760,      18162144000,      6689723040,      1267566300,      128648520,       6621615,      135135;
  8:   1307674368000,    4698287193600,    4250720073600,   1729944216000,    378453675600,    47448601200,    3405402000,   129729600,    2027025;
  9: 355687428096000, 1322395045171200, 1264886951328000, 557287745414400, 136080545000400, 19880055795600, 1777279303800, 95108013000, 2791213425, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-96*n^3+204*n^2-176*n+48+(16*n^3-64*n^2+76*n-24)*x+(4*n^2-8*n+3)*x^2)*P(n-2,x) + (8*n^2-16*n+6+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*P[n - 2, x] + (8*n^2 - 16*n + 6 + (4*n - 2)*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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*v[i - 2] + (8*n^2 - 16*n + 6 + (4*n - 2)*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

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.