Real-rooted atlas

A038284 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 14\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 36 + 168\,x + 196\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
036168196400
12161512352827448000
2129612096423366585638416160000
377769072042336098784011524805378243200000
4466566531843810240118540802074464019361664752953664000000
52799364572288320060161244678402904249604065949443162405121054135041280000000
616796163135283225604812811948912643485099520650551910475897722885059848192147578905625600000000
71007769621163161619752284161075402137637639074816878245079041366159011841366159011847969260902420661046784512000000000

All 15 rows

Technical details
Node id
n-94d6274c5f252185 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5184fe74b2189306

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:        36,        168,          196;
  1:       216,       1512,         3528,         2744;
  2:      1296,      12096,        42336,        65856,         38416;
  3:      7776,      90720,       423360,       987840,       1152480,         537824;
  4:     46656,     653184,      3810240,     11854080,      20744640,       19361664,        7529536;
  5:    279936,    4572288,     32006016,    124467840,     290424960,      406594944,      316240512,      105413504;
  6:   1679616,   31352832,    256048128,   1194891264,    3485099520,     6505519104,     7589772288,     5059848192,     1475789056;
  7:  10077696,  211631616,   1975228416,  10754021376,   37639074816,    87824507904,   136615901184,   136615901184,    79692609024,    20661046784;
  8:  60466176, 1410877440,  14814213120,  92177326080,  376390748160,  1053894094848,  2049238517760,  2732318023680,  2390778270720,  1239662807040,   289254654976;
  9: 362797056, 9311791104, 108637562880, 760462940160, 3548827054080, 11592835043328, 27049948434432, 45083247390720, 52597121955840, 40908872632320, 19090807228416, 4049565169664;
%F Formula
The row polynomials satisfy P(n,x) = (6+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 168*x + 196*x^2.
%t Mathematica
P[0, x_] := 36 + 168*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 14*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] = 36 + 168*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 14*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.