Real-rooted atlas

A013610 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1163652
2256115212962704
34096276486220846656140608
4655365898241990656298598416796167311616
51048576117964805308416011943936013436928060466176380204032
6167772162264924161274019840382205952064497254405804752896217678233619770609664
7268435456422785843228538044416107017666560240789749760325066162176243799621632783641640961028071702528
842949672967730941132860881161420827396522639367705271992320138694895861761560317578444810030613004288282110990745653459728531456
968719476736139156940390412524124635136657516543344642219118333788164993016251023367489524376535047222041363087364062398266736641015599566684162779905883635712
101099511627776247390116249602504824927027201502894956216320591764889010176015977652003274752299580975061401603851755393646592032499186133893120162495930669465603656158440062976144555105949057024
111759218604441643540660459929648983243017420803306368903675904014878660066541568046867779209605939210545250322161336321694772373204500480190661891985506304014299641898912972806434838854510837761316217038422671367516865509350965248
122814749767106567599824371187712940478265934479367053586994508595203570878415969976320128551622974919147523374480103091627622465079259131052818432915177081530430259209151770815304302592061774453003304042496252713671377152901124738381338321616896390877006486250192896
1345035996273704961317302891005870081778358902857924608146714609485778780168252696783575056384033423421973478978355210027026592043693506562256080983209831038976380713665916658987827247589208239582373478404283028741562413613056262822218232239017164898558331837089631436817058172817957820825620325604337285010030592
14720575940379279362269814212194729984331960328533479260162987642956801313341441848604079520812630016831871835784365683507228075674457722341818368721945914627145932472321421331019422193554554882131996529133290331832322398496095274951623311361962405896134051328163841103853316575403872092163821030711222551864934461409422144648154972161056931425538820521590784
Technical details
Node id
n-93e9322de6f5f06b · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-12e1fe8c90bb4450

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:          16,            36;
  2:         256,          1152,           1296;
  3:        4096,         27648,          62208,          46656;
  4:       65536,        589824,        1990656,        2985984,         1679616;
  5:     1048576,      11796480,       53084160,      119439360,       134369280,        60466176;
  6:    16777216,     226492416,     1274019840,     3822059520,      6449725440,      5804752896,      2176782336;
  7:   268435456,    4227858432,    28538044416,   107017666560,    240789749760,    325066162176,    243799621632,     78364164096;
  8:  4294967296,   77309411328,   608811614208,  2739652263936,   7705271992320,  13869489586176,  15603175784448,  10030613004288,   2821109907456;
  9: 68719476736, 1391569403904, 12524124635136, 65751654334464, 221911833378816, 499301625102336, 748952437653504, 722204136308736, 406239826673664, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (16+36*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[(16 + 36*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] = (16 + 36*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.