Real-rooted atlas

A249251 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-4 + 2\,x + 34\,x^{2} + 26\,x^{3} + 4\,n - 9\,n x - 50\,n x^{2} - 39\,n x^{3} + 7\,n^{2} x + 16\,n^{2} x^{2} + 13\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 44\,x^{2} + 92\,x^{3} + 52\,x^{4} - 4\,n x - 24\,n x^{2} - 45\,n x^{3} - 26\,n x^{4}\right)\,P_{n-1}'(x) + \left(4\,x^{2} + 20\,x^{3} + 29\,x^{4} + 13\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
26181337
312578743199
4241563693781421069
5483961275200715484695743
6969603918836498646102154930853
7192225611160301534810245351234215116165751

All 15 rows

Technical details
Node id
n-5e246db2a215203d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-371e17a42c112769

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:   3,     4;
  2:   6,    18,    13;
  3:  12,    57,    87,     43;
  4:  24,   156,   369,    378,    142;
  5:  48,   396,  1275,   2007,   1548,     469;
  6:  96,   960,  3918,   8364,   9864,    6102,    1549;
  7: 192,  2256, 11160,  30153,  48102,   45351,   23421,   5116;
  8: 384,  5184, 30144,  98664, 198945,  253236,  198861,  88146,  16897;
  9: 768, 11712, 78288, 301176, 735195, 1181562, 1250883, 841647, 326769, 55807;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (4*n-4+(7*n^2-9*n+2)*x+(16*n^2-50*n+34)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - ((4*n-4)*x+(24*n-44)*x^2+(45*n-92)*x^3+(26*n-52)*x^4)*d/dx P(n-1,x) + (4*x^2+20*x^3+29*x^4+13*x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (7*n^2 - 9*n + 2)*x + (16*n^2 - 50*n + 34)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - ((4*n - 4)*x + (24*n - 44)*x^2 + (45*n - 92)*x^3 + (26*n - 52)*x^4)*D[P[n - 1, x], x] + (4*x^2 + 20*x^3 + 29*x^4 + 13*x^5)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (7*n^2 - 9*n + 2)*x + (16*n^2 - 50*n + 34)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - ((4*n - 4)*x + (24*n - 44)*x^2 + (45*n - 92)*x^3 + (26*n - 52)*x^4)*deriv(v[i - 1], x) + (4*x^2 + 20*x^3 + 29*x^4 + 13*x^5)*deriv(deriv(v[i - 1], x), x))/(2*n - 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

Candidate equalities ?

Same fingerprint ?

References

  1. [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