Real-rooted atlas

A144445 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 7\,x\right)\,P_{n-2}(x) + \left(-5 - 6\,x + 7\,n + 7\,n x\right)\,P_{n-1}(x) + \left(-7\,x - 7\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
22525151
341462120911245
4969719394113781681142151
52938087345206145261872691345711843581
6109387753281632535650582166072892941922107665199062559
74833627561691769646226034848214214470904138361924519202186132916316817517
8247280719819891228792415685825696112438176314950877042503984881566968638593868906411466299514767
9143761171419064692527138551187504252938011372802854840601690173142817051701904652279784931761036383414955125137139115179186621
1093617857926217468089289631085976164967502310109612413222273071288269846052026818864905256254767187277782518620113230915600187296244100110513675177426079583
11675054905268695437128019789778247867036927182260651117064688086804408629477327550325640630990552200405113113390250111791709199453390060116665387593192234438298034135280088491382354045974058989
12533948700167752885320369220100651731082819392115588835271204251754856800896010780965206960233886610936974750457219421679128285243434464589038622209748815228748348399205326052437548541493520058379249828224070801143634177262557996591
1345966842057795556788298784473375671119122847054436898303994308137217019580828965835213974031138066354247009269813542662604390234012202481373402024121102908382980716474481810194459836174007015770906957914296955789500732439557652828746958662922579256641715419608961463107760541
144278653952276161051479299505776659331273603539239438017178601848156116500877137358545520477718841505067910856604841414331695017574902628669473358110826213397756803249321970088173798366015415856336376399991392699366975505437456784470856984567830859321282389362305131829792654174539280218063405313451727818291181518143616767
Technical details
Node id
n-07cdb61aa05a9163 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9024966cc99eeda3

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:             2,             1;
  2:            25,            25,              1;
  3:           414,           621,            209,              1;
  4:          9697,         19394,          11378,           1681,             1;
  5:        293808,        734520,         614526,         187269,         13457,             1;
  6:      10938775,      32816325,       35650582,       16607289,       2941922,        107665,            1;
  7:     483362756,    1691769646,     2260348482,     1421447090,     413836192,      45192021,       861329,           1;
  8:   24728071981,   98912287924,   156858256961,   124381763149,   50877042503,    9848815669,    686385938,     6890641,        1;
  9: 1437611714190, 6469252713855, 11875042529380, 11372802854840, 6016901731428, 1705170190465, 227978493176, 10363834149, 55125137, 1;
%F Formula
The row polynomials satisfy P(n,x) = (7+7*x)*P(n-2,x) + (7*n-5+(7*n-6)*x)*P(n-1,x) - (7*x+7*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(7 + 7*x)*P[n - 2, x] + (7*n - 5 + (7*n - 6)*x)*P[n - 1, x] - (7*x + 7*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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (7 + 7*x)*v[i - 2] + (7*n - 5 + (7*n - 6)*x)*v[i - 1] - (7*x + 7*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