Real-rooted atlas

A131431 → D+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 18\,x^{2} + 3\,n + 18\,n x + 27\,n x^{2} - 9\,n^{2} - 54\,n^{2} x - 81\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 30\,x - 24\,n - 72\,n x + 18\,n^{2} + 54\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181220
2218463168
3402705402701120
4656242106280810536656
596120057601296012960388836864
613320521282541040692555540413851194560
717632342494810395024948033679822453248114991232
8225480044100226800708750136080015309008748001640254915200
92806804725764445281714608428652068584326613488330674455112423855104
103419300112995803520369117011389896237289503254256027457785122034601830519113770496
1140812342168300136323072705602671930868706792122690700147228840110421630441686526022998534773760
124811598424175821978001335163556966976174461364384527088600823575640878480432592974157306536196633172483027968
1356020280336960339768023166000112586760400308480105080976020265616802814669000270208224016580959205526986406377292011408506880
1464525284457821507124838351313209188980847215369258198969659305075831016658442812708230535110908193766238585899191956489220566766751942260736
Technical details
Node id
n-a0ea580767a9b2ca · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-20d8219123bfe5ea

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:   8,   12;
  2:  21,   84,    63;
  3:  40,  270,   540,    270;
  4:  65,  624,  2106,   2808,    1053;
  5:  96, 1200,  5760,  12960,   12960,    3888;
  6: 133, 2052, 12825,  41040,   69255,   55404,   13851;
  7: 176, 3234, 24948, 103950,  249480,  336798,  224532,   48114;
  8: 225, 4800, 44100, 226800,  708750, 1360800, 1530900,  874800,  164025;
  9: 280, 6804, 72576, 444528, 1714608, 4286520, 6858432, 6613488, 3306744, 551124;
%F Formula
The row polynomials satisfy (9*n^2-21*n+10)*P(n,x) = -(9*n^2-3*n-2+(54*n^2-18*n-12)*x+(81*n^2-27*n-18)*x^2)*P(n-2,x) + (18*n^2-24*n-10+(54*n^2-72*n-30)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 3*n - 2 + (54*n^2 - 18*n - 12)*x + (81*n^2 - 27*n - 18)*x^2)*P[n - 2, x] + (18*n^2 - 24*n - 10 + (54*n^2 - 72*n - 30)*x)*P[n - 1, x])/(9*n^2 - 21*n + 10)]];
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] = 8 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 3*n - 2 + (54*n^2 - 18*n - 12)*x + (81*n^2 - 27*n - 18)*x^2)*v[i - 2] + (18*n^2 - 24*n - 10 + (54*n^2 - 72*n - 30)*x)*v[i - 1])/(9*n^2 - 21*n + 10)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.