Real-rooted atlas

A038322 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(361 - 152\,x + 456\,n x + 64\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6859 + 8664\,x + 3648\,x^{2} + 512\,x^{3} + 8664\,n x + 6080\,n x^{2} + 1024\,n x^{3} + 1216\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1193251
236112165762153
368593465632832819282539
413032187795212476166225921024002980881
5247609920851360395078402957312097280001179648103316067
64704588147541100811259734401123778560554496000134479872128450563474029817
78938717391053827734429950893504373656371202458265600089429114881708392448134217728114116857371
81698356304122883116518475875596876811359153684489341409280004531075153921298378260482040109465613589544963679332384033
93226876977794891266155808185353243799043237358800076831947619737600193703462830087400756084736174429359308823238121881613421772800116831684924307
10613106625780110326006328928044021395402272087871167430656010116746250240007360731587543043515359140249601104719275622402207621578752025501368320001299227607043662828658622089
111164902588982192158135322745952102227462656387202295634249125888030205713804288000256398816966082561469420120624332857721582151270401537976366530560266489298944000271538569871361236950581248113598525733851339
1222133149190661614473225941691609623307861485656281658156067644522496086086284342220800083512757526095462455837964583724646426321041460979302487664652892241920202531867197440003095539696533504282024732524544116135915683843490472362400584497
13420529834622570599207390063315229765233674351779183232143645487081970565122362590248058726400025784563886181974016197028246459714109441083549540143647948843306338528767508481250634279944192000254866101681258496348300544667811842868557117390848107752139522048106383742062173043651
147990066857828841211883973659109116243211601311479777189497634736090512549245747262844900598362122240076207711041382278758465511891947854941388841174882525458622054419199143414420262092866533743693031014400169485957618036899843088264829387931648381518096612982784286620691128647689895604649984003219379832485334146329
Technical details
Node id
n-dd0715c4bad37c11 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3125eed2334c9e3a

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:           19,            32;
  2:          361,          1216,            576;
  3:         6859,         34656,          32832,           8192;
  4:       130321,        877952,        1247616,         622592,         102400;
  5:      2476099,      20851360,       39507840,       29573120,        9728000,        1179648;
  6:     47045881,     475411008,     1125973440,     1123778560,      554496000,      134479872,      12845056;
  7:    893871739,   10538277344,    29950893504,    37365637120,    24582656000,     8942911488,    1708392448,     134217728;
  8:  16983563041,  228831165184,   758755968768,  1135915368448,   934140928000,   453107515392,  129837826048,   20401094656,   1358954496;
  9: 322687697779, 4891266155808, 18535324379904, 32373588000768, 31947619737600, 19370346283008, 7400756084736, 1744293593088, 232381218816, 13421772800;
%F Formula
The row polynomials satisfy (361+(456*n-152)*x+64*n^2*x^2)*P(n,x) = (6859+(8664*n+8664)*x+(1216*n^2+6080*n+3648)*x^2+(512*n^2+1024*n+512)*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[((6859 + (8664*n + 8664)*x + (1216*n^2 + 6080*n + 3648)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*P[n - 1, x])/(361 + (456*n - 152)*x + 64*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] = ((6859 + (8664*n + 8664)*x + (1216*n^2 + 6080*n + 3648)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*v[i - 1])/(361 + (456*n - 152)*x + 64*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

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.