Real-rooted atlas

A013618 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(33 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1089 + 792\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
010897921442025
1359373920414256172891125
211859211724976940896228096207364100625
3391353937115526051749280188179203421440248832184528125
412914679692817748296256158936012419827203387225604926873629859848303765625
5426184429771084833093961183454284327172450208026081637120569053900868976230435831808373669453125
61406408618241409137052579252071988510083787053709824172138804992050076743270491048624128945959731242998169616815125390625
74641148440195315189213077002822093400839276818745915863628810225045016524837181981878272901381378867214047502008321277045637125159780352756680642578125

All 15 rows

Technical details
Node id
n-456ae85670c94210 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-432dc7cc45625337

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:              1089,                792,                144;
  1:             35937,              39204,              14256,               1728;
  2:           1185921,            1724976,             940896,             228096,              20736;
  3:          39135393,           71155260,           51749280,           18817920,            3421440,             248832;
  4:        1291467969,         2817748296,         2561589360,         1241982720,          338722560,           49268736,           2985984;
  5:       42618442977,       108483309396,       118345428432,        71724502080,        26081637120,         5690539008,         689762304,          35831808;
  6:     1406408618241,      4091370525792,      5207198851008,      3787053709824,      1721388049920,       500767432704,       91048624128,        9459597312,        429981696;
  7:    46411484401953,    151892130770028,    220934008392768,    187459158636288,    102250450165248,     37181981878272,     9013813788672,     1404750200832,     127704563712,      5159780352;
  8:  1531578985264449,   5569378128234360,   9113527846201680,   8837360335710720,   5623774759088640,   2454010803965952,   743639637565440,   154522522091520,   21071253012480,   1702727516160,    61917364224;
  9: 50542106513726817, 202168426054907268, 367578956463467760, 400995225232873920, 291632891078453760, 148467653639940096, 53988237687250944, 14022918879805440, 2549621614510080, 309045044183040, 22476003213312, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (33+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1089 + 792*x + 144*x^2.
%t Mathematica
P[0, x_] := 1089 + 792*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(33 + 12*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] = 1089 + 792*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (33 + 12*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.