Real-rooted atlas

A038321 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 14\,x + 14\,n\right)\,P_{n}(x) = \left(666 + 1288\,x + 392\,x^{2} + 252\,n + 392\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1332 + 2576\,x + 784\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0133225767844692
13304811188810819221952275080
275816037013766265728403916861465615379088
31658620810614240025909632029240064014137088017210368832806816
4351459648278648294489159616001450939392012280826880475006156848189030444076076864
5727861593668886091008273075328512582509491200710960302080481408413696155168677888134929285122292779848832
614815724774416304099696647715242192896203896245288963262053150720031851021533184179725807779844965397692416377801998336117670767448320
7297538935552037335626431488205431656177664648080344203264128454634532044816440747879628801337742904393728647012908007424156410027311104105784559534085974188445116928

All 15 rows

Technical details
Node id
n-d23383cdeaf9b1af · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8ed4d1dc77260f2b

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             1332,              2576,                784;
  1:            33048,            111888,             108192,              21952;
  2:           758160,           3701376,            6265728,            4039168,              614656;
  3:         16586208,         106142400,          259096320,          292400640,           141370880,            17210368;
  4:        351459648,        2786482944,         8915961600,        14509393920,         12280826880,          4750061568,           481890304;
  5:       7278615936,       68886091008,       273075328512,       582509491200,        710960302080,        481408413696,        155168677888,         13492928512;
  6:     148157247744,     1630409969664,      7715242192896,     20389624528896,      32620531507200,      31851021533184,      17972580777984,       4965397692416,        377801998336;
  7:    2975389355520,    37335626431488,    205431656177664,    648080344203264,    1284546345320448,    1644074787962880,    1337742904393728,     647012908007424,     156410027311104,     10578455953408;
  8:   59111068529664,   833109019545600,   5226987700408320,  19173621243248640,   45365624094228480,   71934595337945088,   76723490104934400,   53509716175749120,   22645451780259840,   4866089738567680,    296196766695424;
  9: 1163972315879424, 18206209107136512, 128298789010022400, 536637403908587520, 1476368835730145280, 2794522444204474368, 3692642560681181184, 3375833564617113600, 2060124072766341120, 774977683146670080, 149875563947884544, 8293509467471872;
%F Formula
The row polynomials satisfy (14*n+23+14*x)*P(n,x) = (252*n+666+(392*n+1288)*x+392*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1332 + 2576*x + 784*x^2.
%t Mathematica
P[0, x_] := 1332 + 2576*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((252*n + 666 + (392*n + 1288)*x + 392*x^2)*P[n - 1, x])/(14*n + 23 + 14*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1332 + 2576*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((252*n + 666 + (392*n + 1288)*x + 392*x^2)*v[i - 1])/(14*n + 23 + 14*x)); 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. [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.