Real-rooted atlas

A204420 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 54\,x - 27\,x^{2} + 108\,n - 180\,n x + 72\,n x^{2} - 144\,n^{2} + 168\,n^{2} x - 36\,n^{2} x^{2} + 80\,n^{3} - 48\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 6\,x - 12\,n + 12\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2365490
31035243012154680
45544016254013608034020388080
5478831516159500176053507654500114817547355840
6608855940227132829029301426001700829900454677300454677308011301760
71071108388354313452443306280668043654398076683001597559213252896294401020687817151797118202880

All 15 rows

Technical details
Node id
n-6d28c1bea5c74132 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8b81ab23f79e321a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, 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;
  1:                3;
  2:               36,                54;
  3:             1035,              2430,              1215;
  4:            55440,            162540,            136080,             34020;
  5:          4788315,          16159500,          17605350,           7654500,           1148175;
  6:        608855940,        2271328290,        2930142600,        1700829900,         454677300,         45467730;
  7:     107110838835,      431345244330,      628066804365,      439807668300,      159755921325,      28962944010,       2068781715;
  8:   24921704808000,   106692655469400,   170430834974400,   136640733629400,    60684263640000,   15143482153800,    1986030446400,    106394488200;
  9: 7411923037393875, 33376191321072600, 57416499451010700, 51078468797155800, 26248598960834250, 8090343277216200, 1474999987160700, 146505210251400, 6104383760475;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(48*n^3-168*n^2+180*n-54)*x+(36*n^2-72*n+27)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (12*n - 6)*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] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (12*n - 6)*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.