Real-rooted atlas

A156992 → basis A013611 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-12\,n + 8\,n x - 12\,n^{2} + 8\,n^{2} x\right)\,P_{n-2}(x) + \left(8 + 4\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112820
221638496696
35184157449984153632448
4155520675840752640276480307201891200
555987203174912052531200331776008478720737280132272640
623514624016463462403678474240344236032014811955202890137602064384010793180160
7112870195209406365696026702807040033926086656021411790848069198151680108999475206606028801006516224000
8609499054080589191708672020382406410240331843947724802833292722176013293972357120342456532992045185236992023781703680105595316305920
93656994324480040227123363840016456047132672003295906313011200360703045140480022762657677312008437748465664001802653138944002045226516480095126814720012309092116070400
1024136162541568002976797467607040014079325962240000033657947947008000045424136096317440036640565986590720018158614971678720055406613233664000100977113825280001004539163443200418557984768001578338220146688000
1117378037029928960023749993750855680001276317626510868480035586749468403302400575299264985432064005718194785395671040036086939340138086400146406560376029184003786861511390003200601216689320755200532405756624896002009078326886400220781503063837900800
1213554868883344588800203323058756046028800122445574945835581440039103403129403604992007403334690861062553600881946057794880798720068438479710229364736003522047408471015424000120539224874743234560027016478077306798080037975598534806732800302969011694469120010447207299809280033457031428587179212800
131138608986200945459200185972808221133373440001241083573568016482304004473893608219937341440009743214906542760394752001361244526287323450572800126484380567620152197120079829218701915445002240034528487638396336865280010196103448754714050560020141588841594303283200253908926214564741120018428873676863569920058504360878931968005460052345772172115968000
141024748087580850913280001810388309484440125440000132647939822180987043840005333625281052767630131200013167490883260267678924800021202119421795096539955200023102612514235978717593600017430779590484219972812800092211167566612647641088000342966604518910140088320008892831053635186065408000156935022839691057561600017928661391348701593600011934889619302121472000351026165273591808000954705515838977750335488000
Technical details
Node id
n-e9b21d7b5c1e846e · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              1;
  1:             12,               8;
  2:            216,             384,               96;
  3:           5184,           15744,             9984,             1536;
  4:         155520,          675840,           752640,           276480,            30720;
  5:        5598720,        31749120,         52531200,         33177600,          8478720,           737280;
  6:      235146240,      1646346240,       3678474240,       3442360320,       1481195520,        289013760,        20643840;
  7:    11287019520,     94063656960,     267028070400,     339260866560,     214117908480,      69198151680,     10899947520,       660602880;
  8:   609499054080,   5891917086720,   20382406410240,   33184394772480,   28332927221760,   13293972357120,   3424565329920,    451852369920,    23781703680;
  9: 36569943244800, 402271233638400, 1645604713267200, 3295906313011200, 3607030451404800, 2276265767731200, 843774846566400, 180265313894400, 20452265164800, 951268147200;
%F Formula
The row polynomials satisfy P(n,x) = -(12*n^2+12*n-(8*n^2+8*n)*x)*P(n-2,x) + (8*n+8+(4*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 12 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 12 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(12*n^2 + 12*n - (8*n^2 + 8*n)*x)*P[n - 2, x] + (8*n + 8 + (4*n + 4)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 12 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(12*n^2 + 12*n - (8*n^2 + 8*n)*x)*v[i - 2] + (8*n + 8 + (4*n + 4)*x)*v[i - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

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.