Real-rooted atlas

A178822 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15 + 5\,x + 3\,n + n x\right)\,P_{n-1}(x) + \left(5\,x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118624
218914721357
315122072672564312
4102062205012222226812646872
561236196812166320516606300252482580
633679815555541885422873180175560152464624842222
7173210411244024188274241221343236036005100483326479248164688
884440077590082517124435914967552660837777124864741315314669241287479972493
9394053664854069221449968520166335169089528439025002377437837800308708412612620024824493674
10177324147297154958111602874283171451179901189551363043432118738858459611030629606711705225225300349131440358
11773778096175469417768870679417616650702868814605631040067841757984179139280322787600816257297040136936803843844368508405519440
123288556908100528855700653197959596154097770226416842767021089757863635523245117395526486907627279586466965972967002647845263117661885356020015164
1313660159464561328587288466133352048013706396547480184581894697201314078592226453950043347201347511156992211035032208209439790561303638336488975761002456856857517542256608
14556163635323065751118836323922805431481179353245215601939543381339201677217900077728373004812572425609443013440497921669388062749234548051423876504269827740086768136154652411628630125511347484
Technical details
Node id
n-0c5bef71f8e4beb5 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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:       18,         6;
  2:      189,       147,         21;
  3:     1512,      2072,        672,         56;
  4:    10206,     22050,      12222,       2268,       126;
  5:    61236,    196812,     166320,      51660,      6300,       252;
  6:   336798,   1555554,    1885422,     873180,    175560,     15246,      462;
  7:  1732104,  11244024,   18827424,   12213432,   3603600,    510048,    33264,     792;
  8:  8444007,  75900825,  171244359,  149675526,  60837777,  12486474,  1315314,   66924,   1287;
  9: 39405366, 485406922, 1449968520, 1663351690, 895284390, 250023774, 37837800, 3087084, 126126, 2002;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n+15+(n+5)*x)*P(n-1,x) + (n+5)*x*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[((3*n + 15 + (n + 5)*x)*P[n - 1, x] + (n + 5)*x*D[P[n - 1, x], x])/(n)]];
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] = ((3*n + 15 + (n + 5)*x)*v[i - 1] + (n + 5)*x*deriv(v[i - 1], x))/(n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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.