Real-rooted atlas

A207824 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-30\,n - 18\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-116 - 80\,x - 16\,x^{2} - x^{3} - 104\,n - 78\,n x - 16\,n x^{2} - n x^{3} + 18\,n^{2} + 3\,n^{2} x + 6\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-832 - 664\,x - 190\,x^{2} - 23\,x^{3} - x^{4} + 32\,n + 12\,n x + n x^{2} + 32\,n^{2} + 12\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1718
24714162
3310142211474
4201712542852813585
51298110227315647635126876
68279979140309666340715421200003
7524204589820280146727301113010024911479082
832978414272772239024675339614617517850133756110879674
92063494730270389195076207243468170627126411726824172063179655420
1012850905521062224215374219165778796182548453430392441532383762151701580819651
11797030562144386999411779299515709648351825979604042477663158766955565283026307714219885048
12492534417797749889868816962155477749860017311493504416654968129108410865592104548570510315784130560884677
1330338442481654733442956471535836038777800840157094684104541413086963257784151664712177094171512775917403732911220690067724
141863340887354345131837204671303722203067825042801374534522904442817704410677921618193738116026645873127618492212104211684443559811589553400630
Technical details
Node id
n-cd2e846b820b3c49 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-04af55361b2210c1

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:        7,        1;
  2:       47,       14,        1;
  3:      310,      142,       21,       1;
  4:     2017,     1254,      285,      28,       1;
  5:    12981,    10227,     3156,     476,      35,      1;
  6:    82799,    79140,    30966,    6340,     715,     42,     1;
  7:   524204,   589820,   280146,   72730,   11130,   1002,    49,    1;
  8:  3297841,  4272772,  2390246,  753396,  146175,  17850,  1337,   56,  1;
  9: 20634947, 30270389, 19507620, 7243468, 1706271, 264117, 26824, 1720, 63, 1;
%F Formula
The row polynomials satisfy (2*n^3-30*n-18*n*x-2*n*x^2)*P(n,x) = (6*n^3+18*n^2-104*n-116+(n^3+3*n^2-78*n-80)*x-(16*n+16)*x^2-(n+1)*x^3)*P(n-1,x) + (32*n^2+32*n-832+(12*n^2+12*n-664)*x+(n^2+n-190)*x^2-23*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^3 + 18*n^2 - 104*n - 116 + (n^3 + 3*n^2 - 78*n - 80)*x - (16*n + 16)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (32*n^2 + 32*n - 832 + (12*n^2 + 12*n - 664)*x + (n^2 + n - 190)*x^2 - 23*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 30*n - 18*n*x - 2*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6*n^3 + 18*n^2 - 104*n - 116 + (n^3 + 3*n^2 - 78*n - 80)*x - (16*n + 16)*x^2 - (n + 1)*x^3)*v[i - 1] + (32*n^2 + 32*n - 832 + (12*n^2 + 12*n - 664)*x + (n^2 + n - 190)*x^2 - 23*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 30*n - 18*n*x - 2*n*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