Real-rooted atlas

A013620 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1141832
21965043241024
327441058413608583232768
4384161975683810243265921049761048576
553782434574408890560114307207348320188956833554432
6752953658084992186701760320060160308629440158723712340122241073741824
710541350494872153636593544967841473920100818950407777461888333319795261222003234359738368
81475789056151795445766830795059217564901580828229306112029035857715218665908531268568643584110199605761099511627776
920661046784239077827072122954311065636886293319687113785140224914629518028878396815831044319824545792138851503257619835929036835184372088832
1028925465497637189884211202151700443648073772586639360165988319938560256096265048064274388855408640201591812136960971960522803202777030065152035704672266241125899906842624
1140495651696645727242168524836817985369088014201222928076803651743038648320657313746956697684511767465861127761284767272960498939735038976021383131501670405498519529000966426841007923236028797018963968
125669391237529687470607664742461854215420067842650894946574336076686603811614720157755299269607424236632948904411136260779168180371456209554688716369920119745536409354240461875640436080641079709289331097611568313814261761152921504606846976
137937147732541441326637549581926410234061096774860848246288027652915215507735437459865603588933058383568896615245667151468953679103014348046008327627790669275865088544842190662561792028020455519788892169825354532912988162105433114195640322082296486567116836893488147419103232
1411112006825558016200016122860044288167156331247322726485966113212908830723039516145742133657678158986604797722624150735188452109893632221488440174528823296249174495196344926208213578138739724222464137300232046965571584641923162816981893122063324451911727513640813011136715489283748133675820810241180591620717411303424
Technical details
Node id
n-b3e0081a257e50f9 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-27210632b0078bca

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          14,           18;
  2:         196,          504,           324;
  3:        2744,        10584,         13608,          5832;
  4:       38416,       197568,        381024,        326592,        104976;
  5:      537824,      3457440,       8890560,      11430720,       7348320,       1889568;
  6:     7529536,     58084992,     186701760,     320060160,     308629440,     158723712,      34012224;
  7:   105413504,    948721536,    3659354496,    7841473920,   10081895040,    7777461888,    3333197952,     612220032;
  8:  1475789056,  15179544576,   68307950592,  175649015808,  282293061120,  290358577152,  186659085312,   68568643584,   11019960576;
  9: 20661046784, 239077827072, 1229543110656, 3688629331968, 7113785140224, 9146295180288, 7839681583104, 4319824545792, 1388515032576, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (14+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(14 + 18*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; for(n = 1, N, my(i = n + 1); v[i] = (14 + 18*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

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.