Real-rooted atlas

A038303 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 3\,n x\right)\,P_{n}(x) = \left(121 + 66\,x + 9\,x^{2} + 33\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111617
212113227280
3133121788911084508
4146413194419602475240571344
51610514392303593701306802227514581114064
6177156157978365929605287496073507596228510317210368
71948717174405562913159175534298018866925370477839293117496263533760
82143588819353842081339300116974036448415072350108673488172889641539648590494005713152
923579476911157537957418941530212160716013928218432530268966882857053581276212576584585119683060507351296
10259374246011414768614602604460404152525537361601506712630505917271421615689734830279446112032152180521651300649539909086058496
11285311670611171187002366635015523211353819875259420260446040415011933164033563796915828868453244888012968046135130990365078594219212576413594968783872
1231384283767212054244028399246220490638982560248371381604297359666847522502537891856835321482349222316566504324279455224555763576060051872184542806008486908733202478258483200
13345227121439312447974133842386008663783067668011551710756886828027026213254022328648169261706442456799085318781683529612239241942213206047844145024725741297420062960632987948819223205223004777355890688
1437974983358324128999078200902047711118521603497112161723950596321051516162036840568826512424229563284901729338229117013197037651231414054318346763462735996660951941039949910298986457767607205906334373603887174453544448027302232064
Technical details
Node id
n-fbfec7e38f09f3ea · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3843d29729075732

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         11,           6;
  2:        121,         132,          27;
  3:       1331,        2178,         891,         108;
  4:      14641,       31944,       19602,        4752,        405;
  5:     161051,      439230,      359370,      130680,      22275,       1458;
  6:    1771561,     5797836,     5929605,     2874960,     735075,      96228,      5103;
  7:   19487171,    74405562,    91315917,    55342980,   18866925,    3704778,    392931,    17496;
  8:  214358881,   935384208,  1339300116,   974036448,  415072350,  108673488,  17288964,  1539648,   59049;
  9: 2357947691, 11575379574, 18941530212, 16071601392, 8218432530, 2689668828, 570535812, 76212576, 5845851, 196830;
%F Formula
The row polynomials satisfy (11+3*n*x)*P(n,x) = (121+(33*n+66)*x+(9*n+9)*x^2)*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[((121 + (33*n + 66)*x + (9*n + 9)*x^2)*P[n - 1, x])/(11 + 3*n*x)]];
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] = ((121 + (33*n + 66)*x + (9*n + 9)*x^2)*v[i - 1])/(11 + 3*n*x)); 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.