Real-rooted atlas

A094638 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(1 + 5\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
2669232190
31056226416163845320
4221766444070160339206144196840
55765762118960311296022851208381441228809054640
6178738567952558414735680014554976080825344239247362949120498005200
7643458816336338899275315611529365781760698510566431244001287760609288257536031872332800
8263818114561584896307844164224549766250070090245860943485443516237864961317993021442821953945626424115202326680294400
91213563326976824026822848024861059637120437415307942404946059235788837274090726400187212242124806042873692160113745435033695126814720190787784140800
106189172967577646879621273152015975247706323203225260455991040427215144202188838794023213619202445746063892480105703552679936029972511955353650349641564160380507258880017361688356812800
11346593686184345628975824018699264110088420515596800250905675443320320381151940816831488405221193445230592307655481717903360166806816312033280632941698741698561600748518794854424284682937958401674231939072001736168835681280000
122114221485724508161933890234509140992810623320434896947220589490386791185920352937408111060961284301378595936697753638217601670163173376249426789174933913601186767154530195865640145767504250142729164958549430763521267792929305395208036313307545600189242403089259520000
131395386180578175385613863070720337074790463557368368150731673617804304921543646832643400040393646144012288467418439549573751603247590785801178522746883633532095403048845312208028562569958562201688208098588375279206426924671744834378137656025217790455578624712291991068606464041788829199237120022330603564532623360000
14990724188210504523776106241964725631013171205288905114477578137088016200269122912430224896034110697550952064713523252226935412438142832640059964890533612881240064052448917926021619539968035117807676724131450060817912374503692324306944068513052148322564833280190556066402295978393603643139509924222992384428553583729877975040234017443515727872002835986652695643166720000
Technical details
Node id
n-8431ed683c9ae995 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4a49296f0fcc9fdd

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:             6,             4;
  2:            66,            92,             32;
  3:          1056,          2264,           1616,            384;
  4:         22176,         64440,          70160,          33920,           6144;
  5:        576576,       2118960,        3112960,        2285120,         838144,         122880;
  6:      17873856,      79525584,      147356800,      145549760,       80825344,       23924736,        2949120;
  7:     643458816,    3363388992,     7531561152,     9365781760,     6985105664,     3124400128,      776060928,      82575360;
  8:   26381811456,  158489630784,   416422454976,   625007009024,   586094348544,   351623786496,   131799302144,   28219539456,    2642411520;
  9: 1213563326976, 8240268228480, 24861059637120, 43741530794240, 49460592357888, 37274090726400, 18721224212480, 6042873692160, 1137454350336, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+1+4*n*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[(5*n + 1 + 4*n*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] = (5*n + 1 + 4*n*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

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.