Real-rooted atlas

A130123 → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
182432
2643845761024
35124608138241382432768
44096491522211844423683317761048576
5327684915202949120884736013271040796262433554432
62621444718592353894401415577603185049603822059521911029761073741824
7209715244040192396361728198180864059454259201070176665610701766656458647142434359738368
816777216402653184422785843225367150592951268147202283043553283424565329922935341711361100753141761099511627776
9134217728362387865643486543872304405807104136982613196841094783959048218956791808105672301608967925422620672264180754022435184372088832
1010737418243221225472043486543872034789235097601826434842624065751654334464164379135836160281792804290560317016904826880211344603217920634033809653761125899906842624
11858993459228346784153642520176230403826815860736022960895164416096435759690547228930727907164166199441694392320929916254158848092991625415884805579497524953088152168114316902436028797018963968
126871947673624739011624964081936918118440819369181184027553074197299201322547561470361646289164651462656119029280532332544223179900998123520297573201330831360267815881197748224146081389744226304365203474360565761152921504606846976
135497558138882144047674163238592858134937642452143948431363183910796132352017193118299114700868772473196458803220631741958937640964642141940760969216773690323460161536092842838815219384327596232266699767808379811613334988390487648833846535782436893488147419103232
14439804651110418471795346636836020000925941764322400111113011235659800916682342421395880550009405449628146247504232448330107871343002255368665331622753809203217330663245507618406425995994868261427609628359267129012466483221269450346759349862498166693908120076288210357201231685877761180591620717411303424
Technical details
Node id
n-7549daa79994df99 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f9762973ec1089d7

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:         8,         24;
  2:        64,        384,         576;
  3:       512,       4608,       13824,        13824;
  4:      4096,      49152,      221184,       442368,        331776;
  5:     32768,     491520,     2949120,      8847360,      13271040,       7962624;
  6:    262144,    4718592,    35389440,    141557760,     318504960,     382205952,     191102976;
  7:   2097152,   44040192,   396361728,   1981808640,    5945425920,   10701766656,   10701766656,     4586471424;
  8:  16777216,  402653184,  4227858432,  25367150592,   95126814720,  228304355328,  342456532992,   293534171136,  110075314176;
  9: 134217728, 3623878656, 43486543872, 304405807104, 1369826131968, 4109478395904, 8218956791808, 10567230160896, 7925422620672, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (8+24*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[(8 + 24*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] = (8 + 24*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.