Real-rooted atlas

A225471 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2\,x + 2\,x^{2} + 4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14610
2328860180
3384150419528404680
4614430976583044857615120159120
51228807546881847552225420813710723326406683040
62949120213073926396313610213734491515136436312328648640334152000
7825753606850805762430189568477873971256265441283967128576155109888025945920019380816000
82642411520247301406721010588057602355461160963425265623043182506147841845175357446104005632088216128001279133856000
9951268147209906967019524577862352896123198947983362128170865459224473031499776187361546485769209108385792263710020096033522128640094655905344000
103805072588800436231942963202247237555978246850660107550721368703927713792187275302348390417773135089336321155282830671872492266560241664124167059896320140792940288007761784238208000
11167423193907200209445388812288011894512183934976404801972307230729173600931414016014536151370812620816434843347333939213258886596050944074802738861539328281076124065546246331173692497920647647525324800698560581438720000
1280363133075456001089049463252582406756592792350228482537775076271456256642733830861488128011564153123697065984151568004921266012161458168723977142272010219974763379359744508930233859158835217092769575676313603476457658404864003238237626624000068458936980994560000
134178882919923712006097018127520890880410151496217851330561684499050448069591044712614461666324643849484122310974510202881412617894270224826368157671496304295044710413188497986433835335688165223588292480204803637047280774626017281103785355323573862402045675492123074560017486483183769600007256647319985423360000
14234017443515727872003656705360767274188802650475430216179122176118120733605727274270723616073547793022163353680444248819121940070400134114511482784749453312170227875798078264967168165305056580520603811840122218540415754134093824677257615844342908846082727607221830484452966475475333364656501555201284416091260493004800101421602465863680000827257794478338263040000
Technical details
Node id
n-1a4773d98a3fdb95 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d47399ee7119df41

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:           4,            6;
  2:          32,           88,            60;
  3:         384,         1504,          1952,            840;
  4:        6144,        30976,         58304,          48576,          15120;
  5:      122880,       754688,       1847552,        2254208,        1371072,         332640;
  6:     2949120,     21307392,      63963136,      102137344,       91515136,       43631232,        8648640;
  7:    82575360,    685080576,    2430189568,     4778739712,     5626544128,     3967128576,     1551098880,     259459200;
  8:  2642411520,  24730140672,  101058805760,   235546116096,   342526562304,   318250614784,   184517535744,   61040056320,    8821612800;
  9: 95126814720, 990696701952, 4577862352896, 12319894798336, 21281708654592, 24473031499776, 18736154648576, 9209108385792, 2637100200960, 335221286400;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n+(8*n+2)*x+(4*n+2)*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[Together[((4*n + (8*n + 2)*x + (4*n + 2)*x^2)*P[n - 1, x])/(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] = ((4*n + (8*n + 2)*x + (4*n + 2)*x^2)*v[i - 1])/(1 + x)); 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. [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.