Real-rooted atlas

A110441 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-32\,n - 8\,n x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-64 - 40\,x - 8\,x^{2} - 64\,n - 48\,n x - 8\,n x^{2} + n x^{3} - 8\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3} + 4\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-64 - 96\,x - 40\,x^{2} - 4\,x^{3} + 8\,n x^{2} + 8\,n x^{3} + n x^{4} + 8\,n^{2} x^{2} + 8\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21416333
34888364176
4164416270645919
55601816163260810064722
619127552871244801150144723957
765283040042912283201000019441968120308
822288119552199824161536731001948830382569599091
976096461984892800855936474320161784344964480324102962230
10259808176102438645284291072282100011773443206565683263184001114558993
118870406639744163140482060512015699200777100825872005848328856086004841271185848
12302854424811008674867529559040082911200475683841877366451824649995401320001137457613346495919
131034009692021632274535424431133696419734400274224960125416704410158089651312162140018972814688676141679900538
143530329633914470411012006401899401216205203232015054443527840486082970316808274895216945280251970426457618590784158116104717
Technical details
Node id
n-d2044edb5cd72a9a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-252adc5b3ad97b1c

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:     4,      2;
  2:    14,     16,      3;
  3:    48,     88,     36,      4;
  4:   164,    416,    270,     64,      5;
  5:   560,   1816,   1632,    608,    100,      6;
  6:  1912,   7552,   8712,   4480,   1150,    144,     7;
  7:  6528,  30400,  42912,  28320,  10000,   1944,   196,    8;
  8: 22288, 119552, 199824, 161536,  73100,  19488,  3038,  256,   9;
  9: 76096, 461984, 892800, 855936, 474320, 161784, 34496, 4480, 324, 10;
%F Formula
The row polynomials satisfy -(32*n+8*n*x-2*n^3*x^2)*P(n,x) = -(64*n+64+(8*n^2+48*n+40)*x-(4*n^3+4*n^2-8*n-8)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (64+96*x-(8*n^2+8*n-40)*x^2-(8*n^2+8*n-4)*x^3-(n^2+n)*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[(-(64*n + 64 + (8*n^2 + 48*n + 40)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (64 + 96*x - (8*n^2 + 8*n - 40)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-32*n - 8*n*x + 2*n^3*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] = (-(64*n + 64 + (8*n^2 + 48*n + 40)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (64 + 96*x - (8*n^2 + 8*n - 40)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-32*n - 8*n*x + 2*n^3*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. [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