Real-rooted atlas

A038323 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(20\,n + 27\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 27\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12727
2108014582538
3324008748059049178929
486400034992004723920212576411212884
52160000011664000023619600021257640071744535658756935
651840000034992000009447840000127545840008609344200232452293437153891134
71209600000097977600000330674400000595213920000602654094000325433210760732224724212037271697181
8276480000000261273600000010581580800000238085568000003214155168000026034656860800117155955873602259436291848109430594020008
9622080000000067184640000000317447424000000857108044800000144636982560000015620794116480001054403602862400406698532532640686303773648835786142658807923
10138240000000000167961600000000090699264000000002857026816000000057854793024000000781039705824000007029357352416000040669853253264000137260754729766002058911320946490302165227737747090
11304128000000000041057280000000000249422976000000000897922713600000000212134241088000000034365747056256000003866146543828800000298245590523936000015098683020274260004529604906082278006114966623211075315621942274041524553
12663552000000000009853747200000000006651279360000000000269376814080000000007273173980160000000013746298822502400000018557503410378240000017894735431436160000012078946416219408000054355258872987336000146759198957065807201801135623563989452800979585687219986172
1314376960000000000002329067520000000000017293326336000000000077819968512000000000023637815435520000000005105768134072320000000804158481116390400000093052624243468032000007851315170542615200000471078910232556912000019078695864418554936004682952621266372575205268321698924669147140783210571240950962591
143096576000000000000054344908800000000000044019376128000000000002178959118336000000000073539870243840000000000178701884692531200000000321663392446556160000000434245579802850816000000439673649550386451200000329755237162789838400000178067828067906512736000655613366977292160528001475130075698907361188015318658478411730289262064257888913580441029606
Technical details
Node id
n-9d80e6ff5fccd1bf · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-94e7d97c3019c560

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:            27;
  2:          1080,           1458;
  3:         32400,          87480,           59049;
  4:        864000,        3499200,         4723920,         2125764;
  5:      21600000,      116640000,       236196000,       212576400,         71744535;
  6:     518400000,     3499200000,      9447840000,     12754584000,       8609344200,       2324522934;
  7:   12096000000,    97977600000,    330674400000,    595213920000,     602654094000,     325433210760,      73222472421;
  8:  276480000000,  2612736000000,  10581580800000,  23808556800000,   32141551680000,   26034656860800,   11715595587360,   2259436291848;
  9: 6220800000000, 67184640000000, 317447424000000, 857108044800000, 1446369825600000, 1562079411648000, 1054403602862400, 406698532532640, 68630377364883;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (20*n+27*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 27.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 27;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 27*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] = 27; for(n = 2, N, my(i = n + 1); v[i] = ((20*n + 27*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.