Real-rooted atlas

A038314 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 22\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 1056\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
057610564842116
11382438016348481064897336
23317761216512167270410222082342564477456
37962624364953606690816061332480281107205153632205962976
419110297610510663682408693760294395904020239718407421230081133799049474296896
54586471424294298583048093211033612364627968011334242304062338332672190478238722494357888435817657216
6110075314176807218970624258982753075247480171397125440436305920398965329100818285910917124789167144965487587353620047612231936
72641807540224217949122068487991467809177617092861702963223502684841574421544127771443213165855860326451723005165568118531886837761207269217792922190162669056

All 15 rows

Technical details
Node id
n-c9b6b72f63c86d9a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-60cc3cc9b4461a16

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:              576,              1056,               484;
  1:            13824,             38016,             34848,              10648;
  2:           331776,           1216512,           1672704,            1022208,             234256;
  3:          7962624,          36495360,          66908160,           61332480,           28110720,            5153632;
  4:        191102976,        1051066368,        2408693760,         2943959040,         2023971840,          742123008,          113379904;
  5:       4586471424,       29429858304,       80932110336,       123646279680,       113342423040,        62338332672,        19047823872,         2494357888;
  6:     110075314176,      807218970624,     2589827530752,      4748017139712,      5440436305920,      3989653291008,      1828591091712,       478916714496,        54875873536;
  7:    2641807540224,    21794912206848,    79914678091776,    170928617029632,    235026848415744,    215441277714432,    131658558603264,     51723005165568,     11853188683776,     1207269217792;
  8:   63403380965376,   581197658849280,  2397440342753280,   5860409726730240,   9401073936629760,  10341181330292736,   7899513516195840,   4137840413245440,   1422382642053120,   289744612270080,   26559922791424;
  9: 1521681143169024, 15343618193620992, 70324916720762880, 193393520982097920, 354554788467179520, 455011978532880384, 417094313655140352, 273097467274199040, 125169672500674560, 38246288819650560, 7011819616935936, 584318301411328;
%F Formula
The row polynomials satisfy P(n,x) = (24+22*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 1056*x + 484*x^2.
%t Mathematica
P[0, x_] := 576 + 1056*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 22*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] = 576 + 1056*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 22*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. [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.