Real-rooted atlas

A008517 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21520641
310521013024469
4945252023809241206889
5103953465044100264327308720123605
61351355405408662507053203036606422450402620169
72027025945945018288270188588401109878036788406233764032064074901
8344594251837836004162158005200595403894490601773314404732437666369603628801775623081
965472907539283744501019998980014980405440136426290007934927000292052560864753603276998240362880054989743445
101374931057591662070500268438920750453247114320486591585480345240896000162831789120499528629609418945536967524480399168001882140936521
1131623414322523190503836507562120816250144337207014001785693529620014972984666640863723564704034045544332808905771274401454105802241309673664047900160070552399533589
127905853580625632468286450002272669375977004839085133883006781037759496006565983992568004485941859599202167522214457607301723653224016575803322624237760975296019006033536062270208002874543652787689
132134580466768751849969737866250726284415606750017068210823664000267262826547717002936344785454080023209044289551120133069281481136005505619701582000161267471738755232220802228492841082721413120294431034240087178291200126484802362553045
146190283353629375577759779672075002460459751362112506326074294730190001094862336960892500134615353222350000012091673193141141608033513936286238403950635048436076001423658883793910403676277205377712065392333865648647484595398438404850381813760013076743680005977683917752887689
Technical details
Node id
n-ad5f496fdc5dd47c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0c4bf0a352e1721b

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,          2;
  2:        15,         20,           6;
  3:       105,        210,         130,          24;
  4:       945,       2520,        2380,         924,         120;
  5:     10395,      34650,       44100,       26432,        7308,        720;
  6:    135135,     540540,      866250,      705320,      303660,      64224,       5040;
  7:   2027025,    9459450,    18288270,    18858840,    11098780,    3678840,     623376,     40320;
  8:  34459425,  183783600,   416215800,   520059540,   389449060,  177331440,   47324376,   6636960,   362880;
  9: 654729075, 3928374450, 10199989800, 14980405440, 13642629000, 7934927000, 2920525608, 647536032, 76998240, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+1+2*n*x)*P(n-1,x) - (x+x^2)*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[(2*n + 1 + 2*n*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], 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] = (2*n + 1 + 2*n*x)*v[i - 1] - (x + x^2)*deriv(v[i - 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. [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