Real-rooted atlas

A136124 → basis A008277 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 6\,x + 5\,x^{2} + x^{3} + n + n x - n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
212322165
360240315136751
4360192037803264104510369
52520168004410057120365759276166391
620160161280529200913920877800445248932893040897
7181440169344066679201439424018433800140253125877207104737662320735
81814400193536008890560023030784036867600037400832023508828083790080129755611414919681
919958400239500800125737920038000793607299784800925670592077579132404147608960128458053917572114035239232359
102395008003193344000188606880006514421760014599569600022216094208023273739720016590435840077074832340210865368002581284541954978797761
1131135104004566481920029967537600011644528896002982483504000529483578624066562895599205931080812800367390034154015076873812003691236893634086429218427969171962447
12435891456006974263296005034546316800217364539392006263215358400012707605886976018637610767776019928431531008015430381434468084430493347200310063899064926865201086912693347907421880179897765505
13653837184000113331778560008924877561600042386085181440013570299943200003097478934950400519190585673760064767402475776006017848759442520411598655067600020154153439219806693571059739201352028419470951254854032087629614610818338391
1410461394944000195279372288000166597714483200086313482551296003039747187276800077092809047654400145373363988652800207255687922483200224666353685854080184396197470284800112863259259630880499786639127193601514271829807464028108730318762242412533676791851060721718024871169
Technical details
Node id
n-a8a1d2c84f050aa7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-21668cd305a13e34

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:        3,         4;
  2:       12,        32,         21;
  3:       60,       240,        315,        136;
  4:      360,      1920,       3780,       3264,       1045;
  5:     2520,     16800,      44100,      57120,      36575,       9276;
  6:    20160,    161280,     529200,     913920,     877800,     445248,      93289;
  7:   181440,   1693440,    6667920,   14394240,   18433800,   14025312,    5877207,    1047376;
  8:  1814400,  19353600,   88905600,  230307840,  368676000,  374008320,  235088280,   83790080,   12975561;
  9: 19958400, 239500800, 1257379200, 3800079360, 7299784800, 9256705920, 7757913240, 4147608960, 1284580539, 175721140;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+2+(n+6)*x-(n-5)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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 + 2 + (n + 6)*x - (n - 5)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], 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] = ((n + 2 + (n + 6)*x - (n - 5)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.