Real-rooted atlas

A038326 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(11\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 12\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11212
2264288552
343569504518419044
46388820908822809682944584016
5878460383328062726404561920124416016790460
611595672632491201379980801505433608211456017915904463416696
7148811124974036448265646304038639462403161410560137952460825082265612435014676
818707684161428586790446753749504850068172809273470976060699082752220723937283439853568326863242912
9231507591482020429889287714368668161683134982144229518406656020030697308161092583489536340545503232464380231688457586410348
1028295372292027780910977601212257933568030857474672640504940494643205508441759744040061394616320187300026777605108182548480619173642240216138319375560
113423740047332373498914254401833540124521605333934907699201018296664197120133304290585804812118571871436807554434413363203090450441830407492001071104081730920775685468299480201668
124108488056798449301856681580826891921826316808800992597703680192021656677171202932694392887705631993029740593152249296335640985601359798194405376049447207069286401078848154238976106993205379072137204605139605488
13489594826768476640924136860550438455448211633024139837993496847360343238711310443520599107568832774144762500542150803456712987519933218816486127854499921920235698353696931840771376430280867841530002836920729613909116699279363418681411395170076
1457998156401804088225193089710396853837627496286233621535050998514493445873195726867589120115328207000309022721677501192731767603218300013011619282944149727379185975951369074386617331875840395973234210845491211781021844289617922142003971689021441797485850368409684678108805326520344
Technical details
Node id
n-d798b01f393943bf · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-60cc3cc9b4461a16

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:          12;
  2:         264,          288;
  3:        4356,         9504,         5184;
  4:       63888,       209088,       228096,         82944;
  5:      878460,      3833280,      6272640,       4561920,       1244160;
  6:    11595672,     63249120,    137998080,     150543360,      82114560,      17915904;
  7:   148811124,    974036448,   2656463040,    3863946240,    3161410560,    1379524608,     250822656;
  8:  1870768416,  14285867904,  46753749504,   85006817280,   92734709760,   60699082752,   22072393728,   3439853568;
  9: 23150759148, 202042988928, 771436866816, 1683134982144, 2295184066560, 2003069730816, 1092583489536, 340545503232, 46438023168;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (11*n+12*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 12.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 12;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 12*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] = 12; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 12*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.