Real-rooted atlas

A038208 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1121628
2144384256784
3172869129216409621952
42073611059222118419660865536614656
52488321658880442368058982403932160104857617210368
6298598423887872796262401415577601415577607549747216777216481890304
7358318083344302081337720832297271296039636172803170893824140928614426843545613492928512
8429981696458647142421403533312570760888329512681472010146860236867645734912257698037764294967296377801998336
9515978035261917364224330225942528102736959897620547391979522739652263936243524645683213915694039044638564679686871947673610578455953408
1061917364224825564856320495338913792017612050268160410947839590406575165433446473057393704960556627761561602783138807808082463372083201099511627776296196766695424
1174300837068810897456103424726497073561602905988294246407749302117990401446536395358208192871519381094418368716131532801224581075435520544258255749120145135534866432175921860444168293509467471872
12891610044825614265760717209610461557859287044649581270794240139487438123827202975732013308313646289164651462656529019024588144644408491871567872026124396275957760104497585103831042533274790395904281474976710656232218265089212416
131069932053790721854548893237248148363911458979847253346782439014424177822608130048058026774259512115210315870979468820481375449463929176064137544946392917606410188514547623526405433874425399214081975954336508805124391009636686233645035996273704966502111422497947648
141283918464548864239664780049121282077094760425717761107783872227049472406187419816584806410831664528442261504216633290568845230083301078713430022553638512584990016929792342334088800150487042282227258667669913611065344284449308672368844809481643622475660473739824332872057594037927936182059119829942534144
Technical details
Node id
n-adc7c5028d9ebff5 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-c91d5d506c107735

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,          16;
  2:        144,         384,          256;
  3:       1728,        6912,         9216,          4096;
  4:      20736,      110592,       221184,        196608,         65536;
  5:     248832,     1658880,      4423680,       5898240,       3932160,       1048576;
  6:    2985984,    23887872,     79626240,     141557760,     141557760,      75497472,      16777216;
  7:   35831808,   334430208,   1337720832,    2972712960,    3963617280,    3170893824,    1409286144,     268435456;
  8:  429981696,  4586471424,  21403533312,   57076088832,   95126814720,  101468602368,   67645734912,   25769803776,   4294967296;
  9: 5159780352, 61917364224, 330225942528, 1027369598976, 2054739197952, 2739652263936, 2435246456832, 1391569403904, 463856467968, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (12+16*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 + 16*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 + 16*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.