Real-rooted atlas

A139548 → basis A013609 → D

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(8\,n + 16\,n x + 8\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 16 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27Pn(1)
00
1161632
22567687682562048
3307215360307203072015360307298304
43276822937668812811468801146880688128229376327684194304
532768029491201179648027525120412876804128768027525120117964802949120327680167772160
631457283460300817301504051904512010380902401453326336145332633610380902405190451201730150403460300831457286442450944
729360128381681664229008998483969966082099249152037786484736503819796485038197964837786484736209924915208396996608229008998438168166429360128240518168576
826843545640265318402818572288012213813248036641439744080611167436813435194572801727382159360172738215936013435194572808061116743683664143974401221381324802818572288040265318402684354568796093022208
924159191044107062476832856499814416428249907205749887467520149497074155522989941483110446984794734592587309934182405873099341824046984794734592298994148311041494970741555257498874675201642824990720328564998144410706247682415919104316659348799488
102147483648040802189312036721970380802080911654912083236466196480249709398589440582655263375360108207406055424016231110908313601983802444349440198380244434944016231110908313601082074060554240582655263375360249709398589440832364661964802080911654912036721970380804080218931202147483648011258999068426240
11188978561024396854978150439685497815040251341486161920113103668772864038455247382773761025473263540633621974427075870720384552473827737605554646844178432066655762130141184666557621301411845554646844178432038455247382773760219744270758707201025473263540633638455247382773761131036687728640251341486161920396854978150403968549781504188978561024396316767208603648
12164926744166437933151158272417264662740992292085263918694414604263195934720554962001445519361664886004336558084043294581960212488086589163920424961347764860653404160188687080491476582422299382239901777922229938223990177792188687080491476582413477648606534041608086589163920424964043294581960212481664886004336558085549620014455193614604263195934720292085263918694441726466274099237933151158272164926744166413835058055282163712
131429365116108835734127902720042880953483264003287539767050240018081468718776320075942168618860544025314056206286848006870958113135001600154596557545537536002920157198082375680046722515169318010880637125206854336512007433127413300592640074331274133005926400637125206854336512004672251516931801088029201571980823756800154596557545537536006870958113135001600253140562062868480075942168618860544018081468718776320032875397670502400428809534832640035734127902720014293651161088479615345916448342016
14123145302310912332492316239462443224001111130112360200009259417600216120005555650560099415202555599257603645224093705306112010935672281115918336027339180702789795840057716048150334013440010388888667060122419201605555521272928010240214074069503057068032024700854173429661696002470085417342966169600214074069503057068032016055555212729280102401038888866706012241920577160481503340134400273391807027897958400109356722811159183360364522409370530611209941520255559925760216120005555650560036020000925941760043224001111130112332492316239462412314530231091216528282690043758247936
Technical details
Node id
n-442c46c4b4bcba0e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f001db5d15224ada

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • 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:          0;
  1:         16,          16;
  2:        256,         768,          768,           256;
  3:       3072,       15360,        30720,         30720,         15360,           3072;
  4:      32768,      229376,       688128,       1146880,       1146880,         688128,         229376,          32768;
  5:     327680,     2949120,     11796480,      27525120,      41287680,       41287680,       27525120,       11796480,        2949120,         327680;
  6:    3145728,    34603008,    173015040,     519045120,    1038090240,     1453326336,     1453326336,     1038090240,      519045120,      173015040,       34603008,        3145728;
  7:   29360128,   381681664,   2290089984,    8396996608,   20992491520,    37786484736,    50381979648,    50381979648,    37786484736,    20992491520,     8396996608,     2290089984,      381681664,      29360128;
  8:  268435456,  4026531840,  28185722880,  122138132480,  366414397440,   806111674368,  1343519457280,  1727382159360,  1727382159360,  1343519457280,   806111674368,   366414397440,   122138132480,   28185722880,    4026531840,    268435456;
  9: 2415919104, 41070624768, 328564998144, 1642824990720, 5749887467520, 14949707415552, 29899414831104, 46984794734592, 58730993418240, 58730993418240, 46984794734592, 29899414831104, 14949707415552, 5749887467520, 1642824990720, 328564998144, 41070624768, 2415919104;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (8*n+16*n*x+8*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 16 + 16*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 16 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 16*n*x + 8*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 16 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 16*n*x + 8*n*x^2)*v[i - 1])/(n - 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.