Real-rooted atlas

A128813 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(2 + 4\,n + 3\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
2658427176
31625327021874867568
4666251828201877678553614580537328
54064125141501451968068713667211473850065610056956768
63454506251458802200256431753024015962161263806793354294000413343008429601664
7390359206251938625011004123072656904867830449283445442497531461949926123443147190034720812001660631527808
85660208490625323259420870008072170364385011511272620908010253148506640958409052902064207816226275964222050739200374984769600420139776535424
9102449773680312566151236639813751897463054128185031732833435763230340984168352458292441378905923879911646695837373516356971023852066063784909308960050622943896000132764169385193984
102264139998334906251630984456312399500528488875418021572510143770228615179080127716676374902611591102168615990733636466021949748339488712710610775879696000729966838928710500116432770960800000835278574284000051246969382684877824
115999970995587501562547704085288981701125017234304422076122681253734507083910705169750539295842920072466497554495370245985158459423931875527992649050887202554646062524131645873014214594028809050017538811841249667900025267176872091000000165385157708232000023727346824183098432512
12187799092161888798906251633537190841874781775006510612879871998205556251572183440740631988873000256187064597493182985867529676573611323050313839962505868704017751860878059155405491572598040995891270251236346561664037767222574974321313887370400048949446069488705886000643017493169606016000038700126903726288000012978858712828154842584064
136854666863908941160078125647510226817479937449281252822329354253111160452606257515866874908362269503306251364288865103042650860745375178258562316444772600657681517248125365555987027883224431251332200436829107654712987680674004658142725656633256274184531498909031725064456794962277125502901695820001571021262703947866527800018966932195516253748800001056513464471727662400008280511858784362789568632832
14288581474970566422839289062529419400611147221832039350000139216637958806597884707082500405321370092615053100346290000811115418767994367501727949750118021953985969092766990363224012876605491867080875659554452281070126805398918645700955737072680733399098679256412677191681329843999228355660938091611616119836039289140017154789325640316553112454369619215003680005747224822948670062387050000641937581013021727674240000332801741308594213656000006094456728065291013122513764352

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(1 + 2\,n + \frac{3}{2}\,n x + 2\,n^{2} + \frac{3}{2}\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-c533a4f2f617ad07 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-87493a606cb929eb)
Fingerprint
f6-a64f9ce16ea07250

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:                5,                3;
  2:               65,               84,                27;
  3:             1625,             3270,              2187,               486;
  4:            66625,           182820,            187767,             85536,             14580;
  5:          4064125,         14150145,          19680687,          13667211,           4738500,            656100;
  6:        345450625,       1458802200,        2564317530,        2401596216,        1263806793,         354294000,          41334300;
  7:      39035920625,     193862501100,      412307265690,      486783044928,      344544249753,      146194992612,       34431471900,       3472081200;
  8:    5660208490625,   32325942087000,    80721703643850,   115112726209080,   102531485066409,    58409052902064,    20781622627596,    4222050739200,    374984769600;
  9: 1024497736803125, 6615123663981375, 18974630541281850, 31732833435763230, 34098416835245829, 24413789059238799, 11646695837373516, 3569710238520660, 637849093089600, 50622943896000;
%F Formula
The row polynomials satisfy 2*P(n,x) = (4*n^2+4*n+2+(3*n^2+3*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[Together[((4*n^2 + 4*n + 2 + (3*n^2 + 3*n)*x)*P[n - 1, x])/(2)]];
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] = ((4*n^2 + 4*n + 2 + (3*n^2 + 3*n)*x)*v[i - 1])/(2)); 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.