Real-rooted atlas

A038255 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(49 + 42\,x + 9\,x^{2} + 21\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110313
291609160
3784819270271900
465179408491410808122000
5528229775570560245704050243250000
6420175950796879795423360110565145807292800000
73294172882367599833586158565222264046437351030218731000000
8255298337906012810588410079866864369513901066867218574921749606561340000000
91960032346893054911067311728952956900539101332199537506480090247164612590490196833700000000
10149308345958800970201033958236510673117280714717675032346079929976875302057529602686729519683005904940000000000
111129900996049271754147970216008301137354060158805321756047171366550177903439564703384070848730960985134156495390177147430000000000
12850250499494067643585608868915746461164259209960102361865413563398316643228302819930091493197488211652283153394923840354648294212576405314414600000000000
136366992112463315976948011793190499192011529590470398113515272971107984225502253412089058180815768713961004460293377549171598936513240202976125738940669087330159432349000000000000
1447475615099432674136687233269635515908231111046669886880121060699939179953528292957245588957851577124725343490848827857482952520814702428523852071553335055350227244008629212232052204782969520000000000000
Technical details
Node id
n-998d0fa1053e63f9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3150a1751f9b8c12

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:        10,         3;
  2:        91,        60,          9;
  3:       784,       819,        270,        27;
  4:      6517,      9408,       4914,      1080,        81;
  5:     52822,     97755,      70560,     24570,      4050,       243;
  6:    420175,    950796,     879795,    423360,    110565,     14580,      729;
  7:   3294172,   8823675,    9983358,   6158565,   2222640,    464373,    51030,    2187;
  8:  25529833,  79060128,  105884100,  79866864,  36951390,  10668672,  1857492,  174960,   6561;
  9: 196003234, 689305491, 1067311728, 952956900, 539101332, 199537506, 48009024, 7164612, 590490, 19683;
%F Formula
The row polynomials satisfy (3*n+4+3*x)*P(n,x) = (21*n+49+(9*n+42)*x+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[((21*n + 49 + (9*n + 42)*x + 9*x^2)*P[n - 1, x])/(3*n + 4 + 3*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] = ((21*n + 49 + (9*n + 42)*x + 9*x^2)*v[i - 1])/(3*n + 4 + 3*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.