Real-rooted atlas

A201637 → reverse (shift +1) → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
23418
3152511152
4105210130261472
5945220517505465715504
6103952772026775113682037120178416
713513540540546084524755563805707124711320064
82027025675675088288205735730193963032593023436502125637824
934459425126351225186486300142101960596295701360931015615167532810131564275648
1065472907526189163004307428125376552176018891572705483077608834625871500002371272036113879795712
111374931057559580345825108030297375106546244805622398476702199702505046150404305418312903168265173581340831377332365568
12316234143225147575933505029240200489503211227869850214147330897589386523226023243486266036359543220318439621513703198622605188178111234698041088
13790585358062539529267903125849615731464501028112336751507708321818697537150564425975115671842486202281515816580272291994975181144928515820357146896046163691363581406419456
142134580466768751138442915610000263791981140187534870084859610002903837588727075158814459924120057726594905479513816779032056021144158620585195981284116810046707139324384795922093367332752112707452084972544
Technical details
Node id
n-25fb05ce44745668 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e733eac92d45f846

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:        1,         1;
  2:        3,         4,         1;
  3:       15,        25,        11,         1;
  4:      105,       210,       130,        26,        1;
  5:      945,      2205,      1750,       546,       57,        1;
  6:    10395,     27720,     26775,     11368,     2037,      120,       1;
  7:   135135,    405405,    460845,    247555,    63805,     7071,     247,     1;
  8:  2027025,   6756750,   8828820,   5735730,  1939630,   325930,   23436,   502,    1;
  9: 34459425, 126351225, 186486300, 142101960, 59629570, 13609310, 1561516, 75328, 1013, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+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 + 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 + 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