Real-rooted atlas

A038336 → x ↦ x/(x+1) → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(24 + 44\,x + 12\,n + 22\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 528 + 968\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
05289681496
19504348483194476296
215206483635215333129370243458752
3228096016727040459993605622144025768160146996960
4328458243010867201103984640202397184018553075206802794245997475968
545984153650582568962318367744056671211520779229158405714347161617460505216237899880064
663063982088093211033644512660684813601090764802493533306880274288663756816762085007364390069882889244109625344
78513637580812486668451848012278923264293783560519686732539928576098743918952448905152590397444741275473510410865422960128353587193169408
81135151677440187300026777601373533529702405875671210393601615809582858240296231756857344036206103615897602844765284106240130385075521536026559922791424013357738408622080
914984002142208274706705940480226633032400896011079837139599360355478108228812807820518381033881611948014193246208012516967250067456086054149844213760350590980846796806427501315524608499579416482465792
101961542098616323955776565542912362612851841433601994370685127884807312692512135577601876924411448131584344102808765490790445061082100242841604130599192522260480252425506209693696092556018943554355215426003157259059218529854720440549376
11255000472820121656100104020426752565676048872636416345690918755500032014259750398664376320418286011694155038728946673027902760550414059057615275766579216109336850836815872013126126322904072192072193694775972397056240645649253241323523676530752480075776682516315536226902016
12329539072567541767854014562859745288639416019145719808580760743509240053762661820074417350246408784006245577255813122147201526696662532096393653613227721464217654127371818811701329925512973055619710320640404284690745445423513620214234537272271175686176571664166527303688710549782798948761624990597399634154258432
13423693093301125120108747893947288780801295912402871857971209503357621060291788804791276133951230443520175680124911545116262404831203435067490697216010122521482998551937024016238211545643510398976019846703000230957154304018192811083545044058112012128540722363362705408055589144977498745733120156789896090381077708802053201020231180779520910371762415244190842880
14542327159425440153614913996884199604224019139629334722825420801520537219369646686208083629547065330567741440337305839830166623223808103065673281439801540608024294051559196524648857604453909452519362852290560635094496007390628937728069860394560812969183150085821699546734414098595840355770527855991972691968015051830024676583460044803942145958843867096678404818178394142504229273633016149250259522654568448
Technical details
Node id
n-8c6dd2889f1f26f5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-57ca34e86b523c55

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:            528,             968;
  1:           9504,           34848,            31944;
  2:         152064,          836352,          1533312,            937024;
  3:        2280960,        16727040,         45999360,          56221440,          25768160;
  4:       32845824,       301086720,       1103984640,        2023971840,        1855307520,         680279424;
  5:      459841536,      5058256896,      23183677440,       56671211520,       77922915840,       57143471616,        17460505216;
  6:     6306398208,     80932110336,     445126606848,     1360109076480,     2493533306880,     2742886637568,      1676208500736,       439006988288;
  7:    85136375808,   1248666845184,    8012278923264,    29378356051968,    67325399285760,    98743918952448,     90515259039744,     47412754735104,    10865422960128;
  8:  1135151677440,  18730002677760,  137353352970240,   587567121039360,  1615809582858240,  2962317568573440,   3620610361589760,   2844765284106240,  1303850755215360,   265599227914240;
  9: 14984002142208, 274706705940480, 2266330324008960, 11079837139599360, 35547810822881280, 78205183810338816, 119480141932462080, 125169672500674560, 86054149844213760, 35059098084679680, 6427501315524608;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(22*n+44)*x)*P(n-1,x) for n >= 1, with P(0,x) = 528 + 968*x.
%t Mathematica
P[0, x_] := 528 + 968*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (22*n + 44)*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] = 528 + 968*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (22*n + 44)*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.