Real-rooted atlas

A116414 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1352\,n + 130\,n x + 50\,n x^{2} - 260\,n^{3} x^{2} - 50\,n^{3} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(3380 + 1846\,x + 450\,x^{2} + 22\,x^{3} - 10\,x^{4} + 3380\,n + 3029\,n x + 385\,n x^{2} - 199\,n x^{3} - 55\,n x^{4} + 1183\,n^{2} x - 715\,n^{2} x^{2} - 541\,n^{2} x^{3} - 90\,n^{2} x^{4} + n^{2} x^{5} - 650\,n^{3} x^{2} - 346\,n^{3} x^{3} - 45\,n^{3} x^{4} + 3\,n^{3} x^{5} - 26\,n^{4} x^{3} + 5\,n^{4} x^{4} + 3\,n^{4} x^{5} + 5\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(2197 + 2535\,x + 806\,x^{2} - 10\,x^{3} - 51\,x^{4} - 5\,x^{5} - 1183\,n x^{2} - 1235\,n x^{3} - 341\,n x^{4} - 25\,n x^{5} - 1183\,n^{2} x^{2} - 1235\,n^{2} x^{3} - 328\,n^{2} x^{4} - 15\,n^{2} x^{5} + n^{2} x^{6} + 26\,n^{3} x^{4} + 20\,n^{3} x^{5} + 2\,n^{3} x^{6} + 13\,n^{4} x^{4} + 10\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
22240971
39527613516522
440916401269320253663
517609008963038086253624867
675734716064611352009000108049164673
73258523906840027527944098125182521715641069524

All 15 rows

Technical details
Node id
n-5a3919624eaffe04 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-47fb531231568563

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,       4;
  2:     22,      40,        9;
  3:     95,     276,      135,       16;
  4:    409,    1640,     1269,      320,      25;
  5:   1760,    9008,     9630,     3808,     625,      36;
  6:   7573,   47160,    64611,    35200,    9000,    1080,     49;
  7:  32585,  239068,   400275,   279440,   98125,   18252,   1715,    64;
  8: 140206, 1184200,  2345445,  2003200,  900250,  229320,  33271,  2560,   81;
  9: 603275, 5764620, 13190850, 13347360, 7336875, 2388204, 473340, 56064, 3645, 100;
%F Formula
The row polynomials satisfy (1352*n+130*n*x-(260*n^3-50*n)*x^2-50*n^3*x^3+2*n^5*x^4)*P(n,x) = (3380*n+3380+(1183*n^2+3029*n+1846)*x-(650*n^3+715*n^2-385*n-450)*x^2-(26*n^4+346*n^3+541*n^2+199*n-22)*x^3+(5*n^5+5*n^4-45*n^3-90*n^2-55*n-10)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (2197+2535*x-(1183*n^2+1183*n-806)*x^2-(1235*n^2+1235*n+10)*x^3+(13*n^4+26*n^3-328*n^2-341*n-51)*x^4+(10*n^4+20*n^3-15*n^2-25*n-5)*x^5+(n^4+2*n^3+n^2)*x^6)*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[((3380*n + 3380 + (1183*n^2 + 3029*n + 1846)*x - (650*n^3 + 715*n^2 - 385*n - 450)*x^2 - (26*n^4 + 346*n^3 + 541*n^2 + 199*n - 22)*x^3 + (5*n^5 + 5*n^4 - 45*n^3 - 90*n^2 - 55*n - 10)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (2197 + 2535*x - (1183*n^2 + 1183*n - 806)*x^2 - (1235*n^2 + 1235*n + 10)*x^3 + (13*n^4 + 26*n^3 - 328*n^2 - 341*n - 51)*x^4 + (10*n^4 + 20*n^3 - 15*n^2 - 25*n - 5)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(1352*n + 130*n*x - (260*n^3 - 50*n)*x^2 - 50*n^3*x^3 + 2*n^5*x^4)]];
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] = ((3380*n + 3380 + (1183*n^2 + 3029*n + 1846)*x - (650*n^3 + 715*n^2 - 385*n - 450)*x^2 - (26*n^4 + 346*n^3 + 541*n^2 + 199*n - 22)*x^3 + (5*n^5 + 5*n^4 - 45*n^3 - 90*n^2 - 55*n - 10)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (2197 + 2535*x - (1183*n^2 + 1183*n - 806)*x^2 - (1235*n^2 + 1235*n + 10)*x^3 + (13*n^4 + 26*n^3 - 328*n^2 - 341*n - 51)*x^4 + (10*n^4 + 20*n^3 - 15*n^2 - 25*n - 5)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(1352*n + 130*n*x - (260*n^3 - 50*n)*x^2 - 50*n^3*x^3 + 2*n^5*x^4)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023