Real-rooted atlas

A103327 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(45 - 39\,x + 6\,x^{2} - 102\,n + 77\,n x - 10\,n x^{2} + 81\,n^{2} - 46\,n^{2} x + 4\,n^{2} x^{2} - 28\,n^{3} + 8\,n^{3} x + 4\,n^{4}\right)\,P_{n}(x) = \left(48\,x - 44\,x^{2} + 8\,x^{3} + 8\,n - 175\,n x + 161\,n x^{2} - 26\,n x^{3} + 11\,n^{2} + 143\,n^{2} x - 114\,n^{2} x^{2} + 12\,n^{2} x^{3} - 8\,n^{3} - 64\,n^{3} x + 24\,n^{3} x^{2} + 4\,n^{4} + 12\,n^{4} x\right)\,P_{n-1}(x) + \left(-18\,x + 36\,x^{2} - 22\,x^{3} + 4\,x^{4} + 44\,n x - 88\,n x^{2} + 52\,n x^{3} - 8\,n x^{4} - 28\,n^{2} x + 44\,n^{2} x^{2} - 16\,n^{2} x^{3} + 8\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
212214
377483128
43364681204928
510892904165024055888
6286012870137284420420634304
7646144616793654732099756727188416
8130561290643500643447601332801999210088991232
92422532558412597001881152119187032558436708144095046272
104202873837838760008230040786828035000287150086300019801025034752
116906915382401054175130310320410823702745758490852301445136102465264011121634816
121085602990460259578009731990017829600016848972083449520214038002732400159500343212580911104
131643855489640589059902795052006669384758504719205941189802273356804662157548906002393824368132734686208
142411649602190124862400732005820220890390036646514103449083680185732820056656314095173650836035234834854601412717129728
Technical details
Node id
n-4143a4d5284977c6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c682981385f84738

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:    12,      2;
  3:    77,     48,       3;
  4:   336,    468,     120,       4;
  5:  1089,   2904,    1650,     240,       5;
  6:  2860,  12870,   13728,    4420,     420,      6;
  7:  6461,  44616,   79365,   47320,    9975,    672,     7;
  8: 13056, 129064,  350064,  344760,  133280,  19992,  1008,    8;
  9: 24225, 325584, 1259700, 1881152, 1191870, 325584, 36708, 1440, 9;
%F Formula
The row polynomials satisfy (4*n^4-28*n^3+81*n^2-102*n+45+(8*n^3-46*n^2+77*n-39)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (4*n^4-8*n^3+11*n^2+8*n+(12*n^4-64*n^3+143*n^2-175*n+48)*x+(24*n^3-114*n^2+161*n-44)*x^2+(12*n^2-26*n+8)*x^3)*P(n-1,x) + ((8*n^3-28*n^2+44*n-18)*x-(8*n^3-44*n^2+88*n-36)*x^2-(16*n^2-52*n+22)*x^3-(8*n-4)*x^4)*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[((4*n^4 - 8*n^3 + 11*n^2 + 8*n + (12*n^4 - 64*n^3 + 143*n^2 - 175*n + 48)*x + (24*n^3 - 114*n^2 + 161*n - 44)*x^2 + (12*n^2 - 26*n + 8)*x^3)*P[n - 1, x] + ((8*n^3 - 28*n^2 + 44*n - 18)*x - (8*n^3 - 44*n^2 + 88*n - 36)*x^2 - (16*n^2 - 52*n + 22)*x^3 - (8*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 28*n^3 + 81*n^2 - 102*n + 45 + (8*n^3 - 46*n^2 + 77*n - 39)*x + (4*n^2 - 10*n + 6)*x^2)]];
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] = ((4*n^4 - 8*n^3 + 11*n^2 + 8*n + (12*n^4 - 64*n^3 + 143*n^2 - 175*n + 48)*x + (24*n^3 - 114*n^2 + 161*n - 44)*x^2 + (12*n^2 - 26*n + 8)*x^3)*v[i - 1] + ((8*n^3 - 28*n^2 + 44*n - 18)*x - (8*n^3 - 44*n^2 + 88*n - 36)*x^2 - (16*n^2 - 52*n + 22)*x^3 - (8*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 28*n^3 + 81*n^2 - 102*n + 45 + (8*n^3 - 46*n^2 + 77*n - 39)*x + (4*n^2 - 10*n + 6)*x^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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023