Real-rooted atlas

A257624 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 3\,n\right)\,P_{n}(x) = \left(-50 + 20\,x + 10\,x^{2} - 30\,n - 48\,n x - 9\,n x^{2} - 36\,n^{2} x - 54\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-30\,x + 30\,x^{3} - 18\,n x - 45\,n x^{2} + 63\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(20 + 8\,x + 21\,n + 24\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(6\,x - 6\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 10\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151015
22516075260
3125183027455005200
46251814060390362803125118560
5312516637010452301393640415925187503043040
6156251455960157399653902192026233275436788010937586944000
778125123664302168184159030794201128849275433636830432825056250002738736000

All 15 rows

Technical details
Node id
n-eab9f49247beab92 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c7ef0268628dd91a

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:       5,        10;
  2:      25,       160,          75;
  3:     125,      1830,        2745,          500;
  4:     625,     18140,       60390,        36280,          3125;
  5:    3125,    166370,     1045230,      1393640,        415925,         18750;
  6:   15625,   1455960,    15739965,     39021920,      26233275,       4367880,       109375;
  7:   78125,  12366430,   216818415,    903079420,    1128849275,     433636830,     43282505,      625000;
  8:  390625, 102993940,  2811644400,  18424936280,   38380875350,   27637404420,   6560503600,   411975760,    3515625;
  9: 1953125, 846607770, 34944852060, 344172869520, 1113098287950, 1335717945540, 602302521660, 93186272160, 3809734965, 19531250;
%F Formula
The row polynomials satisfy (3*n+2)*P(n,x) = -(30*n+50+(36*n^2+48*n-20)*x+(27*n^3+54*n^2+9*n-10)*x^2)*P(n-2,x) - ((18*n+30)*x+(27*n^2+45*n)*x^2-(27*n^2+63*n+30)*x^3)*d/dx P(n-2,x) + (21*n+20+(18*n^2+24*n+8)*x)*P(n-1,x) + ((9*n+6)*x-(9*n+6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 10*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n + 50 + (36*n^2 + 48*n - 20)*x + (27*n^3 + 54*n^2 + 9*n - 10)*x^2)*P[n - 2, x] - ((18*n + 30)*x + (27*n^2 + 45*n)*x^2 - (27*n^2 + 63*n + 30)*x^3)*D[P[n - 2, x], x] + (21*n + 20 + (18*n^2 + 24*n + 8)*x)*P[n - 1, x] + ((9*n + 6)*x - (9*n + 6)*x^2)*D[P[n - 1, x], x])/(3*n + 2)]];
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] = 5 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n + 50 + (36*n^2 + 48*n - 20)*x + (27*n^3 + 54*n^2 + 9*n - 10)*x^2)*v[i - 2] - ((18*n + 30)*x + (27*n^2 + 45*n)*x^2 - (27*n^2 + 63*n + 30)*x^3)*deriv(v[i - 2], x) + (21*n + 20 + (18*n^2 + 24*n + 8)*x)*v[i - 1] + ((9*n + 6)*x - (9*n + 6)*x^2)*deriv(v[i - 1], x))/(3*n + 2)); 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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.