Real-rooted atlas

A034870 → basis A013609 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 64\,x + 64\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1166464144
2256204861448192409620736
34096491522457606553609830407864322621442985984
4655361048576734003229360128734003201174405121174405126710886416777216429981696
51048576209715201887436801006632960352321536084557168641409286144016106127360120795955205368709120107374182461917364224
6167772164026531844429185024295279001601328755507204252017623049921374453761700807049216212600881152018897856102401133871366144412316860416687194767368916100448256
7268435456751619276897710505984781684047872429926226329617197049053184515911471595521179226220789762063645886382082751527848509442751527848509442001111162552321000555581276163078632557772843980465111041283918464548864

All 15 rows

Technical details
Node id
n-6bdf4eb144359de4 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-90334921cba1a1bb

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,            64,             64;
  2:         256,          2048,           6144,            8192,             4096;
  3:        4096,         49152,         245760,          655360,           983040,            786432,            262144;
  4:       65536,       1048576,        7340032,        29360128,         73400320,         117440512,         117440512,           67108864,           16777216;
  5:     1048576,      20971520,      188743680,      1006632960,       3523215360,        8455716864,       14092861440,        16106127360,        12079595520,          5368709120,          1073741824;
  6:    16777216,     402653184,     4429185024,     29527900160,     132875550720,      425201762304,      992137445376,      1700807049216,      2126008811520,       1889785610240,       1133871366144,        412316860416,         68719476736;
  7:   268435456,    7516192768,    97710505984,    781684047872,    4299262263296,    17197049053184,    51591147159552,    117922622078976,    206364588638208,     275152784850944,     275152784850944,     200111116255232,     100055558127616,      30786325577728,       4398046511104;
  8:  4294967296,  137438953472,  2061584302080,  19241453486080,  125069447659520,   600333348765696,  2201222278807552,   6289206510878720,  14150714649477120,   25156826043514880,   35219556460920832,   38421334321004544,   32017778600837120,   19703248369745920,    8444249301319680,    2251799813685248,    281474976710656;
  9: 68719476736, 2473901162496, 42056319762432, 448600744132608, 3364505580994560, 18841231253569536, 81645335432134656, 279926864338747392, 769798876931555328, 1710664170959011840, 3079195507726221312, 4478829829419958272, 5225301467656617984, 4823355200913801216, 3445253714938429440, 1837468647967162368, 689050742987685888, 162129586585337856, 18014398509481984;
%F Formula
The row polynomials satisfy P(n,x) = (16+64*x+64*x^2)*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 + 64*x + 64*x^2)*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 + 64*x + 64*x^2)*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.