Real-rooted atlas

A253284 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 5\,x + 11\,x^{2} + 11\,x^{3} + 4\,x^{4} + n + 3\,n x - n x^{2} - 7\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 10\,x^{2} + 18\,x^{3} + 14\,x^{4} + 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26303672
3242166005281368
412016807920154801080036000
5720144001008003240004903202836801213920
65040136080131040060984001489320018411120910224049956480
7403201411200177811201114848003880800007645881607993440003450585602427788160
8362880159667202540160002047368960947479680026326218240434773785603938880960015090727680136075645440
936288001959552003832012800384072192002258710272008266696704001910606745600271633098240021715356096007478884224008641341273600
10399168002594592000610727040007424524800005374876953600247955178240007497333043200014830129036800018510238454400013242178857600041422381862400613197730252800
11479001600368831232001027458432000148586296320001293016919040007289580469248002761365667737600710858971468800012288165740928000136693860145920008849378766412800253556910351360048086638196889600
1262270208005604318720001822026286080030858001574400031686818042880002133332491530240098216806431744000315266578224998400706319106869376000108416797269580800010877591454073344006430932379872000001699835823189504004129635803592499200
138717829120090665422848003399953356800006657108672614400794739097152000006276381895719936003433572095592038400133374050271221760003711783138876933120073592466401405952000101574463322645913600928110359461469184005049094847981184000012386182292118835200385458023310629068800
1413076743680001556132497920006663908579328000149223952829952000204593893222118400018683100690718464000119344380741121536000548679644786617344000184141965097548057600045211127108654707200008037814703017042944000100777625434286760960008453569414371105024000425903960111095065600097472352383204198400038854351257533779968000
Technical details
Node id
n-3f9dac393051c6af · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-56b745d77a7c0967

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2,         4;
  2:       6,        30,         36;
  3:      24,       216,        600,         528;
  4:     120,      1680,       7920,       15480,        10800;
  5:     720,     14400,     100800,      324000,       490320,       283680;
  6:    5040,    136080,    1310400,     6098400,     14893200,     18411120,       9102240;
  7:   40320,   1411200,   17781120,   111484800,    388080000,    764588160,     799344000,     345058560;
  8:  362880,  15966720,  254016000,  2047368960,   9474796800,  26326218240,   43477378560,   39388809600,   15090727680;
  9: 3628800, 195955200, 3832012800, 38407219200, 225871027200, 826669670400, 1910606745600, 2716330982400, 2171535609600, 747888422400;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n+1+(3*n+5)*x-(n-11)*x^2-(7*n-11)*x^3-(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx 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[((n + 1 + (3*n + 5)*x - (n - 11)*x^2 - (7*n - 11)*x^3 - (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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] = ((n + 1 + (3*n + 5)*x - (n - 11)*x^2 - (7*n - 11)*x^3 - (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.