Real-rooted atlas

A038327 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 - 39\,x + 117\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 1521\,x + 351\,x^{2} + 27\,x^{3} + 1521\,n x + 585\,n x^{2} + 54\,n x^{3} + 117\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1131225
216931281562
321976084315943211872
42856110545682134224642025240640
53712931713660177957073008013162587484734976
6482680926733096347016151898208051333756823443572191095040
7627485174054519566315693934318423201557123753104665232506111399681721761792
88157307216023857632109472028128982320256404852175010762839361690317721455667253144132078036992
9106044993738809891786818297467557217515524499294735408950314813051286592239108851565312621785971968300590558003200
101378584918491272539924760297333847804532528831212802052600527250818513933328214247771010369011635204041608805255879000714492910763188043776
11179216039403718197320924068472430447067155814528579288041931696485250195079154109846127486250886131921659584019265001970518295348500102172484725509168194476119162880
12232980851224812580710967413287369914974247541007851620409920817668081462375434747829159072159314642523036411595577902087513350768495951358122000796945380663979430208898135293487650883305472
1330287510659225336345012791070361132286936952576617032692384927648153539895296823759184047890985396384631065519901211593275444241922539512559751314019488065450044894589777183362618525761517848640131248730862135601108877312
143937376385699289508830179074985041717301854378074512818136376415301762794426094402192251857218573510380088750356740577752330142516155028992770318809791230714630936558238002042703834861692039988572294413812422624915687269005610761680251100567158938664960
Technical details
Node id
n-093a1aafc679d7e2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cbbf8868a78f08ff

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:          13,          12;
  2:         169,         312,           81;
  3:        2197,        6084,         3159,          432;
  4:       28561,      105456,        82134,        22464,        2025;
  5:      371293,     1713660,      1779570,       730080,      131625,        8748;
  6:     4826809,    26733096,     34701615,     18982080,     5133375,      682344,      35721;
  7:    62748517,   405451956,    631569393,    431842320,   155712375,    31046652,    3250611,    139968;
  8:   815730721,  6023857632,  10947202812,   8982320256,  4048521750,  1076283936,  169031772,  14556672,   531441;
  9: 10604499373, 88098917868, 182974675572, 175155244992, 94735408950, 31481305128, 6592239108, 851565312, 62178597, 1968300;
%F Formula
The row polynomials satisfy (169+(117*n-39)*x+9*n^2*x^2)*P(n,x) = (2197+(1521*n+1521)*x+(117*n^2+585*n+351)*x^2+(27*n^2+54*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2197 + (1521*n + 1521)*x + (117*n^2 + 585*n + 351)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(169 + (117*n - 39)*x + 9*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (1521*n + 1521)*x + (117*n^2 + 585*n + 351)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(169 + (117*n - 39)*x + 9*n^2*x^2)); 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.