Real-rooted atlas

A038208 → basis A013611 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 32\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123244
214476810241936
3172813824368643276885184
420736221184884736157286410485763748096
5248832331776017694720471859206291456033554432164916224
629859844777574431850496011324620802264924160241591910410737418247256313856
735831808668860416535088332823781703680634178764801014686023689019431321634359738368319277809664
84299816969172942848856141332484566087106561522029035520324699527577643293270343683298534883328109951162777614048223625216
9515978035212383472844813209037701128218956791808328758271672328766887244595215585577323724817812088369971211874725579980835184372088832618121839509504
106191736422416511297126401981355655168014089640214528065751654334464021040529387028484675673197117440712483534798848071248353479884804222124650659840112589990684262427197360938418176
117430083706882179491220684829059882942464023247906353971201239888338878464046289164651462656123437772403900416235119566483619840313492755311493120278660226943549440148618787703226368360287970189639681196683881290399744
128916100448256285315214344192418462314371481637196650166353920223179900998123520952234244258660352296250653769360998467714435147282513921128573919121375232013375690893290373120107005527146322984965188146770730811392115292150460684697652654090776777588736
1310699320537907237090977864744965934556458359193658026774259512115238684516173008076801856856776304387686466021574268600451072176057531382934536192352115062765869072384521651944838324551680556428741160879521792404675448117003288576179855754718668128256368934881474191032322316779994178213904384
141283918464548864479329560098242568308379041702871048862270977816395776649899871706535690243466132649101523681281386453059640609472512422538075319042886860898592217574443340267521752750534656770493644823370007128756939915264226618250945521841602561510788339636812277350461981060087664093429761180591620717411303424101938319743841411792896
Technical details
Node id
n-94fce0546865473b · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-10cba46fcb583bff

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:         12,           32;
  2:        144,          768,          1024;
  3:       1728,        13824,         36864,         32768;
  4:      20736,       221184,        884736,       1572864,        1048576;
  5:     248832,      3317760,      17694720,      47185920,       62914560,       33554432;
  6:    2985984,     47775744,     318504960,    1132462080,     2264924160,     2415919104,      1073741824;
  7:   35831808,    668860416,    5350883328,   23781703680,    63417876480,   101468602368,     90194313216,     34359738368;
  8:  429981696,   9172942848,   85614133248,  456608710656,  1522029035520,  3246995275776,   4329327034368,   3298534883328,   1099511627776;
  9: 5159780352, 123834728448, 1320903770112, 8218956791808, 32875827167232, 87668872445952, 155855773237248, 178120883699712, 118747255799808, 35184372088832;
%F Formula
The row polynomials satisfy P(n,x) = (12+32*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[(12 + 32*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] = (12 + 32*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

Candidate equalities ?

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.