Real-rooted atlas

A038301 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(540 + 1408\,x + 484\,x^{2} + 220\,n + 484\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2160 + 5632\,x + 1936\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02160563219369728
16080028512037171285184802816
21568000107008002509056021807104374809662914560
3384000003449600001177088000183997440011993907201649162244764729344
49088000001013760000045534720000103583744000121438310400633278300167256313856352187318272
5209920000002799104000001561190400000467489792000079759482880007480599920640325082860748831927780966425563645345792
6476160000000738918400000049264230400000183179673600000411391016960000561506759475200438861862010880163470238547968140482236252161829587348619264
71064960000000018855936000000014630584320000006502878412800000181347876864000003258216854323200037059446125363200248270424794726408091776808124416618121839509504129478489286901760

All 15 rows

Technical details
Node id
n-540fd34dbc661863 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e1e7aa9d2da54c7d

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:             2160,               5632,                1936;
  1:            60800,             285120,              371712,               85184;
  2:          1568000,           10700800,            25090560,            21807104,              3748096;
  3:         38400000,          344960000,          1177088000,          1839974400,           1199390720,            164916224;
  4:        908800000,        10137600000,         45534720000,        103583744000,         121438310400,          63327830016,            7256313856;
  5:      20992000000,       279910400000,       1561190400000,       4674897920000,        7975948288000,        7480599920640,         3250828607488,          319277809664;
  6:     476160000000,      7389184000000,      49264230400000,     183179673600000,      411391016960000,      561506759475200,       438861862010880,       163470238547968,        14048223625216;
  7:   10649600000000,    188559360000000,    1463058432000000,    6502878412800000,    18134787686400000,    32582168543232000,     37059446125363200,     24827042479472640,      8091776808124416,      618121839509504;
  8:  235520000000000,   4685824000000000,   41483059200000000,  214581903360000000,   715316625408000000,  1595861316403200000,   2389359026503680000,   2329450899308544000,   1365487336370995200,   395597977286082560,    27197360938418176;
  9: 5160960000000000, 113991680000000000, 1133969408000000000, 6692600217600000000, 25964410306560000000, 69242649339494400000, 128732812856524800000, 165207109832540160000, 140931779408166912000, 73432874533729075200, 19146942100646395904, 1196683881290399744;
%F Formula
The row polynomials satisfy (11*n+16+11*x)*P(n,x) = (220*n+540+(484*n+1408)*x+484*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2160 + 5632*x + 1936*x^2.
%t Mathematica
P[0, x_] := 2160 + 5632*x + 1936*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((220*n + 540 + (484*n + 1408)*x + 484*x^2)*P[n - 1, x])/(11*n + 16 + 11*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2160 + 5632*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((220*n + 540 + (484*n + 1408)*x + 484*x^2)*v[i - 1])/(11*n + 16 + 11*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.