Real-rooted atlas

A178820 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(9 + 3\,x + 3\,n + n x\right)\,P_{n-1}(x) + \left(3\,x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112416
2907010170
3540740240201540
42835612533956303513020
513608437363696011480140056107240
66123628282834280415876031920277284880404
72624401703640285264018505205460007728050401207297680
810825659730875219544051918917077997151600830168630858016561534935
94330260533414201593372001827859009838290027475140415800033924013860220530164140
1016888014283004722110503564616328683801132906060413639226843662829815520639210214502864679184796
1164481508146224514873922328481387558572412171359200565347983214928273362323000682144142011411403203236442367126620
122418056557391827625480292617351133071839901238438751557174899732023861157320476978502058519461043918875194694546410455393825001115
1389282088036688142960304662321600895842911600120641761240085887489688035261466240088072624640137931393601368887520852051203195920655205603759316487360
143252418920179283691160189428541188068968026036001134235895520098082918133204896494042440149762824640029118226280036695458800300724424015779400050741609044068036849445108040
Technical details
Node id
n-69f36ca70371cbe2 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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:      12,        4;
  2:      90,       70,        10;
  3:     540,      740,       240,        20;
  4:    2835,     6125,      3395,       630,       35;
  5:   13608,    43736,     36960,     11480,     1400,       56;
  6:   61236,   282828,    342804,    158760,    31920,     2772,      84;
  7:  262440,  1703640,   2852640,   1850520,   546000,    77280,    5040,    120;
  8: 1082565,  9730875,  21954405,  19189170,  7799715,  1600830,  168630,   8580,   165;
  9: 4330260, 53341420, 159337200, 182785900, 98382900, 27475140, 4158000, 339240, 13860, 220;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n+9+(n+3)*x)*P(n-1,x) + (n+3)*x*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[Together[((3*n + 9 + (n + 3)*x)*P[n - 1, x] + (n + 3)*x*D[P[n - 1, x], x])/(n)]];
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] = ((3*n + 9 + (n + 3)*x)*v[i - 1] + (n + 3)*x*deriv(v[i - 1], x))/(n)); 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

Same fingerprint ?

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.