Real-rooted atlas

A038322 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(19\,n + 24\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 24\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12424
291211522064
3259926566441472133128
46584642495232315187213271047632672
5156385207901568014971392012607488039813120410256120
63565582562251946880568912896071862681604538695680114661785621169215792
7790370800859901787008189163537920318591221760301823262720152500174848321052999681061988992232
817162337388815175119375365750571552768121064664268801529237864448011590013288448488000559513688060251340852189173332544
9366844961685637070648759808163891289253888414041151799296653749187051520660630757441536417240478384128150583029792768237762678620162524651259961816
1077445047466960880427908045440444847785117696013111303140311040248424691079577603137996097847296026425230297661440143053878303129604517490893783040634033809653760120622226864842320
1116186014920594642044549253127744011621648386199808039146605090357248086534600726052864013116823689001697281380718283052810240996608685511802880472077798400327680132513066217635840167384925748592645705431330707041736
12335491945626870724661572297131256322944150924503951360111567824507518156802818555566505721856049843930018206449664629607537072081469445680669507417276416035877912678424903680151064895488104857603816376307067912192438244169232678912267636596967712139616
136905542547486422321046734870355836646472720527835247598592306191696148410941440870229031158641623040175877867349957043814425918843609467353825282806250736664134549504221546110789273780224012437676395187299942404713224739228871557121082463098004716912641139434840004965171212467404808745923837112
1414129802443318371824232026229595543789952175851458219780556595281446991175477310423042572010247646651908096058479390893860717068288984916057159759445360641244104493254432983613441178625309398936510791688271054802799554461696041790592687829327806464143967592034627349381123030896674413207355392294500081724360228864577336745758849703841648
Technical details
Node id
n-2782359f01ec0f55 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f3eacc80e11c3732

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:            24;
  2:           912,           1152;
  3:         25992,          65664,           41472;
  4:        658464,        2495232,         3151872,         1327104;
  5:      15638520,       79015680,       149713920,       126074880,        39813120;
  6:     356558256,     2251946880,      5689128960,      7186268160,      4538695680,      1146617856;
  7:    7903708008,    59901787008,    189163537920,    318591221760,    301823262720,    152500174848,     32105299968;
  8:  171623373888,  1517511937536,   5750571552768,  12106466426880,  15292378644480,  11590013288448,   4880005595136,    880602513408;
  9: 3668449616856, 37070648759808, 163891289253888, 414041151799296, 653749187051520, 660630757441536, 417240478384128, 150583029792768, 23776267862016;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (19*n+24*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 24.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 24;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 24*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] = 24; for(n = 2, N, my(i = n + 1); v[i] = ((19*n + 24*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.