Real-rooted atlas

A357367 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(-3 - 6\,x + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
21890108216
32641872432032409696
4540049680168480249480136080609120
5141840160488071668801578528017146080734832049193280
6455112061130160338385600987940800160438320013741358404849891204855515840
71725292802682892800177143500806437128320013903021440017847599616012609717120037829151360566373588480

All 15 rows

Technical details
Node id
n-97cba47fa866c4c1 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-bbe4c1ef283db3b0

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:            2,             6;
  2:           18,            90,            108;
  3:          264,          1872,           4320,            3240;
  4:         5400,         49680,         168480,          249480,           136080;
  5:       141840,       1604880,        7166880,        15785280,         17146080,          7348320;
  6:      4551120,      61130160,      338385600,       987940800,       1604383200,       1374135840,        484989120;
  7:    172529280,    2682892800,    17714350080,     64371283200,     139030214400,     178475996160,     126097171200,      37829151360;
  8:   7545363840,  133323563520,  1022363596800,   4443578092800,   11972764742400,   20478180602880,   21713932880640,   13051057219200,    3404623622400;
  9: 373944211200, 7400106604800, 64623542860800, 326848178764800, 1055097063820800, 2254365226713600, 3188240876620800, 2878041835468800, 1504843641100800, 347271609484800;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-2*n)*P(n-2,x) + (6*n-3+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n)*P[n - 2, x] + (6*n - 3 + (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] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n)*v[i - 2] + (6*n - 3 + (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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.