Real-rooted atlas

A147716 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(15\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
29018108
32025810812916
44050024300486032469984
57593756075001822502430012151574640
6136687501366875054675001093500109350437434012224
723920312528704375014352187538272500574087545927015309714256704
841006250005740875000344452500011481750002296350002755620018370805248814693280768
9691980468751107168750007750181250031000725000775018125012400290001240029007085880177147297538935552
10115330078125020759414062501660753125000775018125000232505437500465010875006200145000531441000265720505904905950778711040
111902946289062538058925781250342530332031251826828437500063938995312501534535887500255755981250292292550002192194125974308501948617117825418478592
123113912109375006850606640625006850606640625004110363984375001644145593750004603607662500092072153250001315316475000131531647500876877650035075106063772922313662762852352
13506010717773437512144257226562500133586829492187508905788632812500400760488476562512824335631250002992344980625005129734252500064121678156255699704725003419822835012435719402072619945116423875620864
1481740192871093750212524501464843750255029401757812500187021561289062500935107806445312503366388103203125089770349418750001795406988375000269311048256250299234498062502393875984500130575053700435250179066961566874564524358189056
Technical details
Node id
n-8f7b77214e515821 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a8e18dd2f0709b33

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:          90,           18;
  3:        2025,          810,          81;
  4:       40500,        24300,        4860,         324;
  5:      759375,       607500,      182250,       24300,       1215;
  6:    13668750,     13668750,     5467500,     1093500,     109350,       4374;
  7:   239203125,    287043750,   143521875,    38272500,    5740875,     459270,     15309;
  8:  4100625000,   5740875000,  3444525000,  1148175000,  229635000,   27556200,   1837080,   52488;
  9: 69198046875, 110716875000, 77501812500, 31000725000, 7750181250, 1240029000, 124002900, 7085880, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (15*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[((15*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] = ((15*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.