Real-rooted atlas

A038323 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(11\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 9\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
199
2198162360
332675346218710800
4479161176129622826244288000
56588452156220264627014434202952457200000
68696754355776305821794047632860194861703188646172800000
711160834354789550211206953451222576740750217545245525742334807834032000000
81403076312803580069619724238072268966882802200638132010803132648294630890434437376892160000000
9173630693611136491812723254499281885325544279445446579376703565033773841458422907483409300303234867844012073600000000
10212215292190156267624249051142131572409763497845640119824746287409803842878060534755066076018751151667603835462841103486784401046080000000000
1125678050354992100931392681077352474003255168769034188920241646571679590237252997649052161763407487990756296450593202320455018865542190091252103451916556991013760000000000
1230813660425988277322943833892113450295204774027846890641171804556763923100840521956594827914442705539576829362495778286957560102100020830082027845460226386045565298552268338915443777222118400000000000
133671961200763573605198269840596162233922142826824424561512986186081452155125427515106628275800559656101781535991443308713792590069862163650075744675431513273002707910660325791884648716248464908460139633044255768277479232000000000000
144349861730135306462667111296209822271274909999575486813824729998726441393736876590648590205259430916077337822391937918117527761832067647845979544112422332936003290251101060425456041016723983411967431637317979514307492508881538831465832027509436945410321920000000000000
Technical details
Node id
n-3890397b83a674f9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6bae7ffb6b8d3725

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:           9;
  2:         198,          162;
  3:        3267,         5346,         2187;
  4:       47916,       117612,        96228,        26244;
  5:      658845,      2156220,      2646270,      1443420,       295245;
  6:     8696754,     35577630,     58217940,     47632860,     19486170,      3188646;
  7:   111608343,    547895502,   1120695345,   1222576740,    750217545,    245525742,     33480783;
  8:  1403076312,   8035800696,  19724238072,  26896688280,  22006381320,  10803132648,   2946308904,   344373768;
  9: 17363069361, 113649181272, 325449928188, 532554427944, 544657937670, 356503377384, 145842290748, 34093003032, 3486784401;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (11*n+9*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 9.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 9;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 9*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] = 9; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 9*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.