Real-rooted atlas

A038320 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(29 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(697 + 696\,x + 144\,x^{2} + 204\,n + 144\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 697 + 696\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
06976961441537
1153172509212528172854665
2319345735216602208200448207361877953
36431117191607002205648012044160300672024883262948009
41263672734630404246897852005293555202167948804329676829859842071626049
524378944691061485093219447697808193139856001111646592036421539846061547523583180867215035273
6463682700492340378690245095128447366223263298564635356544002134361456645827446374483129794564299816962156234538625
78719696801255007773165292126380449272961834246241049616802810906112100123701350403841850621952899091726336112225222656515978035268533758575657

All 15 rows

Technical details
Node id
n-784296607d60c8c4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b9f56a026bd52dff

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:             697,              696,              144;
  1:           15317,            25092,            12528,              1728;
  2:          319345,           735216,           602208,            200448,             20736;
  3:         6431117,         19160700,         22056480,          12044160,           3006720,            248832;
  4:       126367273,        463040424,        689785200,         529355520,         216794880,          43296768,          2985984;
  5:      2437894469,      10614850932,      19447697808,       19313985600,       11116465920,        3642153984,        606154752,         35831808;
  6:     46368270049,     234037869024,     509512844736,      622326329856,      463535654400,      213436145664,      58274463744,       8312979456,        429981696;
  7:    871969680125,    5007773165292,   12638044927296,    18342462410496,    16802810906112,    10012370135040,    3841850621952,     899091726336,     112225222656,      5159780352;
  8:  16246539080089,  104636361615000,  300466389917520,   505521797091840,   550273872314880,   403267461746688,  200247402700800,   65860296376320,   13486375895040,   1496336302080,    61917364224;
  9: 300383091166901, 2144543158571748, 6905999866590000, 13220521156370880, 16682219304030720, 14527230229112832, 8871884158427136, 3776093879500800, 1086694890209280, 197800179793920, 19751639187456, 743008370688;
%F Formula
The row polynomials satisfy (12*n+29+12*x)*P(n,x) = (204*n+697+(144*n+696)*x+144*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 697 + 696*x + 144*x^2.
%t Mathematica
P[0, x_] := 697 + 696*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((204*n + 697 + (144*n + 696)*x + 144*x^2)*P[n - 1, x])/(12*n + 29 + 12*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 697 + 696*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((204*n + 697 + (144*n + 696)*x + 144*x^2)*v[i - 1])/(12*n + 29 + 12*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.