Real-rooted atlas

A207607 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(32 - 64\,n + 32\,n^{2}\right)\,P_{n}(x) = \left(64 + 168\,x + 88\,x^{2} + 30\,x^{3} - 128\,n - 312\,n x - 164\,n x^{2} - 45\,n x^{3} + 64\,n^{2} + 144\,n^{2} x + 76\,n^{2} x^{2} + 15\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(32\,x + 120\,x^{2} + 160\,x^{3} + 60\,x^{4} - 32\,n x - 120\,n x^{2} - 112\,n x^{3} - 30\,n x^{4}\right)\,P_{n-1}'(x) + \left(12\,x^{3} + 36\,x^{4} + 15\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
2818834
316606621163
43216830422255781
564432109612967101443742
61281056345657125004219637717929
7256249610016213602604818108663298785903

All 15 rows

Technical details
Node id
n-c2e960ae82fc763f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-40f50691bbfd11c0

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,     3;
  2:    8,    18,     8;
  3:   16,    60,    66,     21;
  4:   32,   168,   304,    222,     55;
  5:   64,   432,  1096,   1296,    710,    144;
  6:  128,  1056,  3456,   5712,   5004,   2196,    377;
  7:  256,  2496, 10016,  21360,  26048,  18108,   6632,    987;
  8:  512,  5760, 27392,  71712, 112720, 108648,  62584,  19674,  2584;
  9: 1024, 13056, 71808, 223104, 430560, 534096, 425064, 208992, 57558, 6765;
%F Formula
The row polynomials satisfy (32*n^2-64*n+32)*P(n,x) = (64*n^2-128*n+64+(144*n^2-312*n+168)*x+(76*n^2-164*n+88)*x^2+(15*n^2-45*n+30)*x^3)*P(n-1,x) - ((32*n-32)*x+(120*n-120)*x^2+(112*n-160)*x^3+(30*n-60)*x^4)*d/dx P(n-1,x) + (12*x^3+36*x^4+15*x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((64*n^2 - 128*n + 64 + (144*n^2 - 312*n + 168)*x + (76*n^2 - 164*n + 88)*x^2 + (15*n^2 - 45*n + 30)*x^3)*P[n - 1, x] - ((32*n - 32)*x + (120*n - 120)*x^2 + (112*n - 160)*x^3 + (30*n - 60)*x^4)*D[P[n - 1, x], x] + (12*x^3 + 36*x^4 + 15*x^5)*D[P[n - 1, x], {x, 2}])/(32*n^2 - 64*n + 32)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((64*n^2 - 128*n + 64 + (144*n^2 - 312*n + 168)*x + (76*n^2 - 164*n + 88)*x^2 + (15*n^2 - 45*n + 30)*x^3)*v[i - 1] - ((32*n - 32)*x + (120*n - 120)*x^2 + (112*n - 160)*x^3 + (30*n - 60)*x^4)*deriv(v[i - 1], x) + (12*x^3 + 36*x^4 + 15*x^5)*deriv(deriv(v[i - 1], x), x))/(32*n^2 - 64*n + 32)); 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

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023