Real-rooted atlas

A154986 → D+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(3 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-12\,x^{2} + 40\,n x^{2} - 63\,n^{2} x^{2} + 59\,n^{3} x^{2} - 32\,n^{4} x^{2} + 9\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-4}(x) + \left(-8\,x - 4\,x^{2} + 20\,n x + 10\,n x^{2} - 20\,n^{2} x - 12\,n^{2} x^{2} + 10\,n^{3} x + 8\,n^{3} x^{2} - 2\,n^{4} x - 2\,n^{4} x^{2}\right)\,P_{n-3}(x) + \left(-3 - 2\,x - x^{2} + 3\,n - 4\,n x + n x^{2} - n^{2} + 10\,n^{2} x - n^{2} x^{2} - 8\,n^{3} x + 2\,n^{4} x\right)\,P_{n-2}(x) + \left(6 + 4\,x - 6\,n - 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 6\,x + 5\,x^{2}\)
\(P_{3}(x) = 1 + 14\,x + 33\,x^{2} + 12\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
216512
3114331260
412814216425360
51504941256673462520
618214597044926322347720160
7112637593222084975540306209120181440
811848652124984589078827496248812150161771814400
91258181564202723333606894517262705729375363257725019958400
101350353131249112159773627402842010798946637150072300551764846341239500800
111462644933341228662255224995258281349771570100793702417805927784641261204174523113510400
12159611173881719002415083112849401224130071198201941741522473268667517168766442146159421121258543589145600
131754185146185202167873878871403654587410204749361227606744515221549296673542833488022250287609049920520353249742653837184000
14193829529539324220232894247360695634350675411428847304015832964046056601495951860497739302208723470709777100246010805775556749092510461394944000
Technical details
Node id
n-908d7e9158d08282 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-fd45b1527b453234

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: 1,   2;
  2: 1,   6,     5;
  3: 1,  14,    33,     12;
  4: 1,  28,   142,    164,      25;
  5: 1,  50,   494,   1256,     673,      46;
  6: 1,  82,  1459,   7044,    9263,    2234,      77;
  7: 1, 126,  3759,  32220,   84975,   54030,    6209,    120;
  8: 1, 184,  8652, 124984,  589078,  827496,  248812,  15016,   177;
  9: 1, 258, 18156, 420272, 3333606, 8945172, 6270572, 937536, 32577, 250;
%F Formula
The row polynomials satisfy (n^2-3*n+3)*P(n,x) = -(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2*P(n-4,x) - ((2*n^4-10*n^3+20*n^2-20*n+8)*x+(2*n^4-8*n^3+12*n^2-10*n+4)*x^2)*P(n-3,x) - (n^2-3*n+3-(2*n^4-8*n^3+10*n^2-4*n-2)*x+(n^2-n+1)*x^2)*P(n-2,x) + (2*n^2-6*n+6+(2*n^2-4*n+4)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 6*x + 5*x^2, P(3,x) = 1 + 14*x + 33*x^2 + 12*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 6*x + 5*x^2;
P[3, x_] := 1 + 14*x + 33*x^2 + 12*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2*P[n - 4, x] - ((2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x + (2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x^2)*P[n - 3, x] - (n^2 - 3*n + 3 - (2*n^4 - 8*n^3 + 10*n^2 - 4*n - 2)*x + (n^2 - n + 1)*x^2)*P[n - 2, x] + (2*n^2 - 6*n + 6 + (2*n^2 - 4*n + 4)*x)*P[n - 1, x])/(n^2 - 3*n + 3)]];
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] = 1 + 2*x; v[3] = 1 + 6*x + 5*x^2; v[4] = 1 + 14*x + 33*x^2 + 12*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2*v[i - 4] - ((2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x + (2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x^2)*v[i - 3] - (n^2 - 3*n + 3 - (2*n^4 - 8*n^3 + 10*n^2 - 4*n - 2)*x + (n^2 - n + 1)*x^2)*v[i - 2] + (2*n^2 - 6*n + 6 + (2*n^2 - 4*n + 4)*x)*v[i - 1])/(n^2 - 3*n + 3)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.