Real-rooted atlas

A081580 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(25 + 60\,x\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
161218
261204144409
35162616403217288892
4462130648714246739220736194821
5406263357001054080154656010368002488324262598
6359281354517213977360278985602934144015178752298598493286549
7317133636412464172900224437028480629544960511598592214990848358318082041478712

All 15 rows

Technical details
Node id
n-3c83a3830f60f25e · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-4db3a90e791dd224

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:         6,         12;
  2:        61,        204,         144;
  3:       516,       2616,        4032,        1728;
  4:      4621,      30648,       71424,       67392,        20736;
  5:     40626,     335700,     1054080,     1546560,      1036800,       248832;
  6:    359281,    3545172,    13977360,    27898560,     29341440,     15178752,      2985984;
  7:   3171336,   36412464,   172900224,   437028480,    629544960,    511598592,    214990848,     35831808;
  8:  28010041,  366716976,  2036495232,  6233079168,  11429061120,  12764086272,   8414502912,   2974040064,   429981696;
  9: 247343646, 3637014108, 23126828544, 83136143232, 185331649536, 264295913472, 239726739456, 132613521408, 40418279424, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (25+60*x)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 12*x;
P[n_, x_] := P[n, x] = Expand[(25 + 60*x)*P[n - 2, x] + (6 + 12*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] = 6 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (25 + 60*x)*v[i - 2] + (6 + 12*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. [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