Real-rooted atlas

A136124 → basis A008277 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28210
36030393
4480360724916
542004200126014059805
640320504002016033602406114486
742336063504031752070560756037871454425
8483840084672005080320141120020160015120560820014408
95987520011975040083825280279417604989600498960277207929296909721
1079833600017962560001437004800558835200119750400149688001108800475201080104726308610
111141620480028540512000256864608001141620480028540512004281076803963960022651207722014301180383524661
12174356582400479480601600479480601600239740300800684972288001198701504013318905609513504043243201201201848121454973802140
1328332944640008499883392000934987173120051943731840001669619952000333923990400432864432003710266560210810600780780018018023401327928182224293
14488198430720001586644899840001903973879808001163539593216004155498547200093498717312001385166182400138516618240944431488043723680013453440262080291214566674115632366
Technical details
Node id
n-fd387c50289c7223 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a76d8ad579b60b88

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        1;
  2:        8,         2;
  3:       60,        30,        3;
  4:      480,       360,       72,        4;
  5:     4200,      4200,     1260,      140,       5;
  6:    40320,     50400,    20160,     3360,     240,      6;
  7:   423360,    635040,   317520,    70560,    7560,    378,     7;
  8:  4838400,   8467200,  5080320,  1411200,  201600,  15120,   560,   8;
  9: 59875200, 119750400, 83825280, 27941760, 4989600, 498960, 27720, 792, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n^2+2*n+n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + n*x)*P[n - 1, x] + 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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 2*n + n*x)*v[i - 1] + 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

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.