Real-rooted atlas

A225470 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 18\,x - 9\,x^{2} + 18\,n - 18\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 6\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
218542799
31625944861081350
4194481008505324040522194
52916013316416402582620182251458428652
65248802571912355168821432606378759185451039526572
71102248057159432861590525921521221432600422528442865217496239660208
8264539520143796124823250346921765801296736439445178564176250761421889568590496735365136
9714256704040412190816692467623005697677376026505619875747241475413020304501369936807971615196830209203521120
102142770112001255221126720225925772767219862902620001008831497850319592674170652143651398573914800701502120324769506495397117993208160
117071141369600427079592489608020400008341674584609556688407400279115501417835163987032706484091825065312974485208653241034100797501286087222125764263318896863360
12254561089305600157991338118016030795298196232963005861944374432174633329065338065708475951690016736757347009729448778396752358465829564729638464570015883177167497428776690873310523998066976640
1399278824829184006314398840185984012721127318061926412954673510909603279378980626885940319131054675112688827132023642933172233403794223223919130061927123507862167790159751324032371381029356188608410922320522451935801295937280
1441697106428257280027116148077756236805627021421394378368594980796730451040038197174429883196001626114759892262640482435414129292624102602482246063800158589120040644151784634546444750144671501247273822622838310031120387798570309644307174453520754082122599750400
Technical details
Node id
n-05453a33d5800878 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-aa53210b1406856d

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          3,           6;
  2:         18,          54,          27;
  3:        162,         594,         486,         108;
  4:       1944,        8100,        8505,        3240,         405;
  5:      29160,      133164,      164025,       82620,       18225,       1458;
  6:     524880,     2571912,     3551688,     2143260,      637875,      91854,       5103;
  7:   11022480,    57159432,    86159052,    59215212,    21432600,    4225284,     428652,     17496;
  8:  264539520,  1437961248,  2325034692,  1765801296,   736439445,  178564176,   25076142,   1889568,   59049;
  9: 7142567040, 40412190816, 69246762300, 56976773760, 26505619875, 7472414754, 1302030450, 136993680, 7971615, 196830;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(18*n-18)*x+9*x^2)*P(n-2,x) + (6*n-3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (18*n - 18)*x + 9*x^2)*P[n - 2, x] + (6*n - 3 + 6*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] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (18*n - 18)*x + 9*x^2)*v[i - 2] + (6*n - 3 + 6*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.