Real-rooted atlas

A038320 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 36 + 132\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
036132121289
12161188217813314913
21296950426136319441464183521
37776712802613604791604392301610511419857
4466565132162352240574992079061405797836177156124137569
5279936359251219758816603741601106859601217545567440556219487171410338673
61679616246343681580705285795919361328231520194807289617857334889353842082143588816975757441
7100776961662819841219401216521632742414344900416262989840963214320278425255373616115753795742357947691118587876497

All 15 rows

Technical details
Node id
n-eef41637184d467b · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-57ca34e86b523c55

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,        132,         121;
  1:       216,       1188,        2178,         1331;
  2:      1296,       9504,       26136,        31944,         14641;
  3:      7776,      71280,      261360,       479160,        439230,        161051;
  4:     46656,     513216,     2352240,      5749920,       7906140,       5797836,       1771561;
  5:    279936,    3592512,    19758816,     60374160,     110685960,     121754556,      74405562,      19487171;
  6:   1679616,   24634368,   158070528,    579591936,    1328231520,    1948072896,    1785733488,     935384208,     214358881;
  7:  10077696,  166281984,  1219401216,   5216327424,   14344900416,   26298984096,   32143202784,   25255373616,   11575379574,    2357947691;
  8:  60466176, 1108546560,  9145509120,  44711377920,  143449004160,  315587809152,  482148041760,  505107472320,  347261387220,  141476861460,   25937424601;
  9: 362797056, 7316407296, 67067066880, 368868867840, 1352519182080, 3471465900672, 6364354151232, 8334273293280, 7639750518840, 4668736428180, 1711870023666, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (6+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 132*x + 121*x^2.
%t Mathematica
P[0, x_] := 36 + 132*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 11*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 + 132*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 11*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

Candidate equalities ?

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.