Real-rooted atlas

A038221 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2361854
332432481729
42592388819443248748
51944038880291609720121598415
61399683499203499201749604374043741062882
79797762939328367416024494409185401837081530911160261
8671846423514624352719362939328014696640440899273483252488114791256
945349632181398528317447424317447424198404640793618561984046428343521771471162261467
1030233088013604889602720977920317447424023808556801190427840396809280850305601062882059049011622614670
11199538380899769190402244806784029930757120261894124801571364748865473531201870672320350751060389723401948617115063885233
12130606940167183381708817958454272026937681408026937681408018856376985694281884928336721017608418025440140300424014030042463772921129718145924
1384894511104509367066624140075943321623345990553602626423937280210113914982412256645040645252847874561641514960803647811024054717165364974287762072619911014751922759
14548549148672356556946636810696708399104196106320650242451329008128022061961073152147079740487687353987024384275774513414476604031504015320806300820892008592174100071666961566106758364789818
Technical details
Node id
n-a4d5926477bcae59 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-11aec07ce60b6d61

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        3;
  2:       36,        18;
  3:      324,       324,        81;
  4:     2592,      3888,      1944,       324;
  5:    19440,     38880,     29160,      9720,      1215;
  6:   139968,    349920,    349920,    174960,     43740,     4374;
  7:   979776,   2939328,   3674160,   2449440,    918540,   183708,    15309;
  8:  6718464,  23514624,  35271936,  29393280,  14696640,  4408992,   734832,   52488;
  9: 45349632, 181398528, 317447424, 317447424, 198404640, 79361856, 19840464, 2834352, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (6*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 3*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 3*n*x)*v[i - 1])/(n - 1)); 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.