Real-rooted atlas

A028421 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-8 + 12\,x - 6\,x^{2} + x^{3} + 12\,n - 12\,n x + 3\,n x^{2} - 6\,n^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n-3}(x) + \left(-7 + 9\,x - 3\,x^{2} + 9\,n - 6\,n x - 3\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 3\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 8 + 12\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2812323
32858304120
4124310225605724
56681898169564010565012
64248132721369264401505168739332
731176104928120624662761946031082528345832
8259488927912116178071811624832550148583836093371976
9241603290893521220226082812203244815776538114870102004951036135792
102488608097778736139174728101976600441463001203879621208112402401683066011422379792
1128100448011465955361715850840134094690463016470019081464638478594521162446728026510858125349561984
12345188736014562243072227628573121880019627294681386803126010602702823121109374408117644678565704018310921372996193152
134583253888019918582041632355518832028040908750414987208236053236325142130986445592280506800282729447247647401494493589681365141067779243008
1465410958592029198720816644908341518272443838915520024985090312009451717635362510266718964797726648066738092856755949204916611725006808417516801516670798231040
Technical details
Node id
n-616ef1e4acab58b1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8a6c98e635141472

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,       2;
  2:       8,      12,        3;
  3:      28,      58,       30,       4;
  4:     124,     310,      225,      60,       5;
  5:     668,    1898,     1695,     640,     105,      6;
  6:    4248,   13272,    13692,    6440,    1505,    168,      7;
  7:   31176,  104928,   120624,   66276,   19460,   3108,    252,     8;
  8:  259488,  927912,  1161780,  718116,  248325,  50148,   5838,   360,   9;
  9: 2416032, 9089352, 12202260, 8281220, 3244815, 776538, 114870, 10200, 495, 10;
%F Formula
The row polynomials satisfy P(n,x) = (n^3-6*n^2+12*n-8+(3*n^2-12*n+12)*x+(3*n-6)*x^2+x^3)*P(n-3,x) - (3*n^2-9*n+7+(6*n-9)*x+3*x^2)*P(n-2,x) + (3*n-3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 8 + 12*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 8 + 12*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[(n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*P[n - 2, x] + (3*n - 3 + 3*x)*P[n - 1, 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] = 3 + 2*x; v[3] = 8 + 12*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*v[i - 3] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*v[i - 2] + (3*n - 3 + 3*x)*v[i - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.