Real-rooted atlas

A204420 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 36\,x - 12\,x^{2} + 108\,n - 120\,n x + 32\,n x^{2} - 144\,n^{2} + 112\,n^{2} x - 16\,n^{2} x^{2} + 80\,n^{3} - 32\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 4\,x - 12\,n + 8\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
233481293
391517709001203705
44798510920076440201601680255465
5408523510374210880740032508005292003024027077085
651395998514153832001372930020625363200143866800159667206652804088135205
78966869411526311622337028194836670014890687812042756714000675026352054486432017297280833709301425

All 15 rows

Technical details
Node id
n-6577090cee20b8ba · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-42e9f6f4286ebba9

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:                3,                 2;
  2:               33,                48,                12;
  3:              915,              1770,               900,               120;
  4:            47985,            109200,             76440,             20160,             1680;
  5:          4085235,          10374210,           8807400,           3250800,           529200,            30240;
  6:        513959985,        1415383200,        1372930020,         625363200,        143866800,         15966720,          665280;
  7:      89668694115,      263116223370,      281948366700,      148906878120,      42756714000,       6750263520,       544864320,       17297280;
  8:   20723300222625,    64033427858400,    74207504170800,    43823437257600,   14739033909600,    2927737612800,    337815878400,    20756736000,    518918400;
  9: 6128733996259875, 19775144834294850, 24429227538315600, 15760339461266400, 5978999145319200, 1401584457873600, 204705525024000, 18049019788800, 873339667200, 17643225600;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(32*n^3-112*n^2+120*n-36)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(8*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (8*n - 4)*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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (8*n - 4)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.