Real-rooted atlas

A249250 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 + 4\,x + 30\,x^{2} + 24\,x^{3} + n - 8\,n x - 42\,n x^{2} - 36\,n x^{3} + 4\,n^{2} x + 12\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x + 31\,x^{2} + 78\,x^{3} + 48\,x^{4} - n x - 14\,n x^{2} - 36\,n x^{3} - 24\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 13\,x^{3} + 24\,x^{4} + 12\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
23121025
33244828103
433612616876409
53482525645642081639
63604261392228018245686553
73726482844669686405760155226215

All 15 rows

Technical details
Node id
n-2c4c65d10f103b9b · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-aecab2aaf9adbf77

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: 3, 12,   10;
  3: 3, 24,   48,   28;
  4: 3, 36,  126,  168,    76;
  5: 3, 48,  252,  564,   564,   208;
  6: 3, 60,  426, 1392,  2280,  1824,    568;
  7: 3, 72,  648, 2844,  6696,  8640,   5760,   1552;
  8: 3, 84,  918, 5112, 16020, 29376,  31248,  17856,  4240;
  9: 3, 96, 1236, 8388, 33228, 80496, 120672, 109152, 54576, 11584;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n-1+(4*n^2-8*n+4)*x+(12*n^2-42*n+30)*x^2+(12*n^2-36*n+24)*x^3)*P(n-1,x) - ((n-1)*x+(14*n-31)*x^2+(36*n-78)*x^3+(24*n-48)*x^4)*d/dx P(n-1,x) + (x^2+13*x^3+24*x^4+12*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[((n - 1 + (4*n^2 - 8*n + 4)*x + (12*n^2 - 42*n + 30)*x^2 + (12*n^2 - 36*n + 24)*x^3)*P[n - 1, x] - ((n - 1)*x + (14*n - 31)*x^2 + (36*n - 78)*x^3 + (24*n - 48)*x^4)*D[P[n - 1, x], x] + (x^2 + 13*x^3 + 24*x^4 + 12*x^5)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
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] = ((n - 1 + (4*n^2 - 8*n + 4)*x + (12*n^2 - 42*n + 30)*x^2 + (12*n^2 - 36*n + 24)*x^3)*v[i - 1] - ((n - 1)*x + (14*n - 31)*x^2 + (36*n - 78)*x^3 + (24*n - 48)*x^4)*deriv(v[i - 1], x) + (x^2 + 13*x^3 + 24*x^4 + 12*x^5)*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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