Real-rooted atlas

A201637 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
211
3325
41520641
510521013024469
6945252023809241206889
7103953465044100264327308720123605
81351355405408662507053203036606422450402620169
92027025945945018288270188588401109878036788406233764032064074901
10344594251837836004162158005200595403894490601773314404732437666369603628801775623081
1165472907539283744501019998980014980405440136426290007934927000292052560864753603276998240362880054989743445
121374931057591662070500268438920750453247114320486591585480345240896000162831789120499528629609418945536967524480399168001882140936521
1331623414322523190503836507562120816250144337207014001785693529620014972984666640863723564704034045544332808905771274401454105802241309673664047900160070552399533589
147905853580625632468286450002272669375977004839085133883006781037759496006565983992568004485941859599202167522214457607301723653224016575803322624237760975296019006033536062270208002874543652787689
152134580466768751849969737866250726284415606750017068210823664000267262826547717002936344785454080023209044289551120133069281481136005505619701582000161267471738755232220802228492841082721413120294431034240087178291200126484802362553045
Technical details
Node id
n-984b9e7687210001 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0c4bf0a352e1721b

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        1;
   3:        3,         2;
   4:       15,        20,         6;
   5:      105,       210,       130,        24;
   6:      945,      2520,      2380,       924,       120;
   7:    10395,     34650,     44100,     26432,      7308,       720;
   8:   135135,    540540,    866250,    705320,    303660,     64224,     5040;
   9:  2027025,   9459450,  18288270,  18858840,  11098780,   3678840,   623376,   40320;
  10: 34459425, 183783600, 416215800, 520059540, 389449060, 177331440, 47324376, 6636960, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-3+(2*n-4)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 3 + (2*n - 4)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 3 + (2*n - 4)*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, v[i])))};
T(10)

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