Real-rooted atlas

A013610 → 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

\(P_{n}(x) = \left(-64 + 96\,x\right)\,P_{n-2}(x) + \left(20 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 16 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1161228
2256528144928
340961440010368172830592
465536327936421632172800207361008640
510485766816768130867209400320265420824883233255424
6167772161342218243480330243744645121811496963881779229859841096450048
72684354562550153216838818201612320391168884912947231890309125494210563583180836150575104
84294967296472447057921889766604803565114490883491822960641848762040325279219712075963432964299816961191904804864
968719476736858993721344405444073881694147093463041187819657625685331300843523545342410752835884417024103195607040515978035239297772158976
101099511627776153931638374408383772491776023227250245632036241787584512033489038396620818767382419865663843682222080127962552729601382821134336619173642241295669662842880
11175921860444162726788878827521684451654369280543818766090240010179235062743040116410087353876488364439665377280381581580356812810956896773079041907880382955521832753981030474300837068842718957920059392
122814749767106564785074620858368330733090756362241221601545053798402679463693674086403683301269679636483271192502442393601906202835167477767301361634482585618103976847212544278479537333862424073471210291289161004482561408468082652872704
1345035996273704968331659317346304063725934432013516826537461475369287686695444212843806720108141409787266990081154455922851341926482966106697136865284043909906892324864133121870765280460829008631876183654439944130008186880313846735778611210699320537907246437985298333827072
147205759403792793614411518810269941761208766138724594483256078822865496965120160332645220321591296298777873461981216768375084936488281964544323670674014809882624194064201945949667328810019456866456698882333610574050911846445318517595716976645646388091871559684065741804404736012839184645488641531086508191574392832
Technical details
Node id
n-28a2c17d8d00ccc8 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-52999a0df2a941cc

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:          16,           12;
  2:         256,          528,           144;
  3:        4096,        14400,         10368,          1728;
  4:       65536,       327936,        421632,        172800,          20736;
  5:     1048576,      6816768,      13086720,       9400320,        2654208,        248832;
  6:    16777216,    134221824,     348033024,     374464512,      181149696,      38817792,       2985984;
  7:   268435456,   2550153216,    8388182016,   12320391168,     8849129472,    3189030912,     549421056,     35831808;
  8:  4294967296,  47244705792,  188976660480,  356511449088,   349182296064,  184876204032,   52792197120,   7596343296,    429981696;
  9: 68719476736, 858993721344, 4054440738816, 9414709346304, 11878196576256, 8533130084352, 3545342410752, 835884417024, 103195607040, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = -(64-96*x)*P(n-2,x) + (20+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 16 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 16 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(64 - 96*x)*P[n - 2, x] + (20 + 12*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] = 16 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(64 - 96*x)*v[i - 2] + (20 + 12*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. [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.