Real-rooted atlas

A207824 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(256\,n + 24\,n x + 9\,n x^{2} - 80\,n^{3} x^{2} - 15\,n^{3} x^{3} + n^{5} x^{4}\right)\,P_{n}(x) = \left(768 + 648\,x + 147\,x^{2} - 12\,x^{3} - 6\,x^{4} + 768\,n + 1096\,n x + 123\,n x^{2} - 148\,n x^{3} - 33\,n x^{4} + 448\,n^{2} x - 264\,n^{2} x^{2} - 301\,n^{2} x^{3} - 54\,n^{2} x^{4} + n^{2} x^{5} - 240\,n^{3} x^{2} - 181\,n^{3} x^{3} - 27\,n^{3} x^{4} + 3\,n^{3} x^{5} - 16\,n^{4} x^{3} + 3\,n^{4} x^{4} + 3\,n^{4} x^{5} + 3\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(512 + 576\,x + 152\,x^{2} - 42\,x^{3} - 25\,x^{4} - 3\,x^{5} - 448\,n x^{2} - 456\,n x^{3} - 146\,n x^{4} - 15\,n x^{5} - 448\,n^{2} x^{2} - 456\,n^{2} x^{3} - 138\,n^{2} x^{4} - 9\,n^{2} x^{5} + n^{2} x^{6} + 16\,n^{3} x^{4} + 12\,n^{3} x^{5} + 2\,n^{3} x^{6} + 8\,n^{4} x^{4} + 6\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16814
2359636167
32048486481281828
4118966247668307240018953
56930484087516845568120001152189226
6403913392646611765376002140004147231361837039
72354162310304541857655308802952000864000131712819217451080

All 15 rows

Technical details
Node id
n-bb67441be2b035d5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e0e9333fb1293ed5

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:       6,         8;
  2:      35,        96,        36;
  3:     204,       848,       648,       128;
  4:    1189,      6624,      7668,      3072,       400;
  5:    6930,     48408,     75168,     45568,     12000,      1152;
  6:   40391,    339264,    661176,    537600,    214000,     41472,     3136;
  7:  235416,   2310304,   5418576,   5530880,   2952000,    864000,   131712,     8192;
  8: 1372105,  15405888,  42246648,  51913728,  34782000,  13644288,  3139136,   393216,   20736;
  9: 7997214, 101101864, 317387808, 456161792, 367970400, 181173888, 55845888, 10551296, 1119744, 51200;
%F Formula
The row polynomials satisfy (256*n+24*n*x-(80*n^3-9*n)*x^2-15*n^3*x^3+n^5*x^4)*P(n,x) = (768*n+768+(448*n^2+1096*n+648)*x-(240*n^3+264*n^2-123*n-147)*x^2-(16*n^4+181*n^3+301*n^2+148*n+12)*x^3+(3*n^5+3*n^4-27*n^3-54*n^2-33*n-6)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (512+576*x-(448*n^2+448*n-152)*x^2-(456*n^2+456*n+42)*x^3+(8*n^4+16*n^3-138*n^2-146*n-25)*x^4+(6*n^4+12*n^3-9*n^2-15*n-3)*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[((768*n + 768 + (448*n^2 + 1096*n + 648)*x - (240*n^3 + 264*n^2 - 123*n - 147)*x^2 - (16*n^4 + 181*n^3 + 301*n^2 + 148*n + 12)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (512 + 576*x - (448*n^2 + 448*n - 152)*x^2 - (456*n^2 + 456*n + 42)*x^3 + (8*n^4 + 16*n^3 - 138*n^2 - 146*n - 25)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(256*n + 24*n*x - (80*n^3 - 9*n)*x^2 - 15*n^3*x^3 + 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] = ((768*n + 768 + (448*n^2 + 1096*n + 648)*x - (240*n^3 + 264*n^2 - 123*n - 147)*x^2 - (16*n^4 + 181*n^3 + 301*n^2 + 148*n + 12)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (512 + 576*x - (448*n^2 + 448*n - 152)*x^2 - (456*n^2 + 456*n + 42)*x^3 + (8*n^4 + 16*n^3 - 138*n^2 - 146*n - 25)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(256*n + 24*n*x - (80*n^3 - 9*n)*x^2 - 15*n^3*x^3 + 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