Real-rooted atlas

A085880 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-30 - 36\,x + 20\,n + 24\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
25611
35012072242
46252250270010806655
5875042000756006048018144204974
61312507875001890000226800013608003265926764142
7206250014850000445500007128000064152000307929606158592233846052
83351562528153125010135125002027025000243243000017513496007005398401200925448359996359

All 15 rows

Technical details
Node id
n-9a00fc9238e55143 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

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 -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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          1;
   2:          5,            6;
   3:         50,          120,           72;
   4:        625,         2250,         2700,          1080;
   5:       8750,        42000,        75600,         60480,         18144;
   6:     131250,       787500,      1890000,       2268000,       1360800,        326592;
   7:    2062500,     14850000,     44550000,      71280000,      64152000,      30792960,       6158592;
   8:   33515625,    281531250,   1013512500,    2027025000,    2432430000,    1751349600,     700539840,     120092544;
   9:  558593750,   5362500000,  22522500000,   54054000000,   81081000000,   77837760000,   46702656000,   16012339200,   2401850880;
  10: 9496093750, 102557812500, 492277500000, 1378377000000, 2481078600000, 2977294320000, 2381835456000, 1224943948800, 367483184640, 48997757952;
%F Formula
The row polynomials satisfy n*P(n,x) = (20*n-30+(24*n-36)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((20*n - 30 + (24*n - 36)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((20*n - 30 + (24*n - 36)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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