Real-rooted atlas

A094638 → D+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n x - n x^{2} + 4\,n^{2} x + 2\,n^{2} x^{2} - n^{3} - 3\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-n x - 2\,n x^{2} + 4\,n^{2} x + 4\,n^{2} x^{2} - 3\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-2}'(x) + \left(-n x^{2} + 2\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}''(x) + \left(-1 - 2\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,n x + 2\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
21313228
370157656298
444620171661386244534
53276285424133617861267412093809
6272524487521071146759131202504212047202530709
725329678327092965113132198266135062492447488189468504086083647
826022241510760618852623071412312386869944489241002504316130841883664403203595737039
929288160320073408028577263280651336310755644432726922330093689438858875643607516420622096362880180560986449
1035845776074019300030996578475730318243867612537690365527092039850854329561306456486823108702046410924136246500640362880010712560696949
11474057792018571902675753745235219322516523858069758026197105007539318835938663301369529321105789140670792540981598017355980100194075576319413744039916800740181146290389
126737553216050275705323915151199881442712591224357593915201906712887518276117832384299149477857463542350238522936217434121663554120999173483219775909456816602098181124459658688047900160058841863140340017
1310242087206401461015070941510653504791760709705350895929406580351457472397560137903174520287857720438910701200082932111123675148731282289937445461169229445091174877869823566740794478675282908942315545666741292224062270208005326488210945416382
14165836268864004537096813679220030136154522653550003329734521734345252511713717230215327821917740858134380071525113651235222592076633258923306101606450671151569125875925353762405107028337352120238585491283909472146687510567983205375989520720641065768230016087178291200544153582396936955310
Technical details
Node id
n-3e4aca5db862e0c7 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-053c648299955a32

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,          1;
  2:       13,         13,           2;
  3:       70,        157,          65,           6;
  4:      446,       2017,        1661,         386,          24;
  5:     3276,      28542,       41336,       17861,        2674,         120;
  6:    27252,     448752,     1071146,      759131,      202504,       21204,        720;
  7:   253296,    7832709,    29651131,    32198266,    13506249,     2447488,     189468,      5040;
  8:  2602224,  151076061,   885262307,  1412312386,   869944489,   241002504,   31613084,   1883664,    40320;
  9: 29288160, 3200734080, 28577263280, 65133631075, 56444327269, 22330093689, 4388588756, 436075164, 20622096, 362880;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(3*n^3-4*n^2+n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) - ((3*n^3-4*n^2+n)*x+(2*n^3-4*n^2+2*n)*x^2)*d/dx P(n-2,x) - (n^3-2*n^2+n)*x^2*d^2/dx^2 P(n-2,x) + (2*n^2-1+(2*n^2-2*n)*x)*P(n-1,x) + (2*n^2-2*n)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (3*n^3 - 4*n^2 + n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] - ((3*n^3 - 4*n^2 + n)*x + (2*n^3 - 4*n^2 + 2*n)*x^2)*D[P[n - 2, x], x] - (n^3 - 2*n^2 + n)*x^2*D[P[n - 2, x], {x, 2}] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*P[n - 1, x] + (2*n^2 - 2*n)*x*D[P[n - 1, x], x])/(n - 1)]];
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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (3*n^3 - 4*n^2 + n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] - ((3*n^3 - 4*n^2 + n)*x + (2*n^3 - 4*n^2 + 2*n)*x^2)*deriv(v[i - 2], x) - (n^3 - 2*n^2 + n)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*v[i - 1] + (2*n^2 - 2*n)*x*deriv(v[i - 1], x))/(n - 1)); 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

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.