Real-rooted atlas

A081580 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(36\,x^{2} + 48\,x^{3} + n - 5\,n x - 60\,n x^{2} - 72\,n x^{3} + 7\,n^{2} x + 24\,n^{2} x^{2} + 24\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(37\,x^{2} + 132\,x^{3} + 96\,x^{4} - n x - 26\,n x^{2} - 72\,n x^{3} - 48\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 25\,x^{3} + 48\,x^{4} + 24\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
219919
3116422887
4123124202101451
51302557408553422223
6137435198539653567118911179
71446644280129101947214308408855767

All 15 rows

Technical details
Node id
n-a4508f583538f67c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b123934b80f387c2

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: 1,  2;
  2: 1,  9,    9;
  3: 1, 16,   42,    28;
  4: 1, 23,  124,   202,   101;
  5: 1, 30,  255,   740,   855,    342;
  6: 1, 37,  435,  1985,  3965,   3567,   1189;
  7: 1, 44,  664,  4280, 12910,  19472,  14308,   4088;
  8: 1, 51,  942,  7968, 33570,  75042,  90912,  56484,  14121;
  9: 1, 58, 1269, 13392, 74226, 228132, 402906, 407208, 219069, 48682;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+(7*n^2-5*n)*x+(24*n^2-60*n+36)*x^2+(24*n^2-72*n+48)*x^3)*P(n-1,x) - (n*x+(26*n-37)*x^2+(72*n-132)*x^3+(48*n-96)*x^4)*d/dx P(n-1,x) + (x^2+25*x^3+48*x^4+24*x^5)*d^2/dx^2 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[((n + (7*n^2 - 5*n)*x + (24*n^2 - 60*n + 36)*x^2 + (24*n^2 - 72*n + 48)*x^3)*P[n - 1, x] - (n*x + (26*n - 37)*x^2 + (72*n - 132)*x^3 + (48*n - 96)*x^4)*D[P[n - 1, x], x] + (x^2 + 25*x^3 + 48*x^4 + 24*x^5)*D[P[n - 1, x], {x, 2}])/(n)]];
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] = ((n + (7*n^2 - 5*n)*x + (24*n^2 - 60*n + 36)*x^2 + (24*n^2 - 72*n + 48)*x^3)*v[i - 1] - (n*x + (26*n - 37)*x^2 + (72*n - 132)*x^3 + (48*n - 96)*x^4)*deriv(v[i - 1], x) + (x^2 + 25*x^3 + 48*x^4 + 24*x^5)*deriv(deriv(v[i - 1], x), x))/(n)); 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