Real-rooted atlas

A081577 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(3\,n\right)\,P_{n}(x) = \left(18\,x^{2} + 8\,x^{3} + 9\,n - 6\,n x - 30\,n x^{2} - 12\,n x^{3} + 12\,n^{2} x + 12\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(57\,x^{2} + 66\,x^{3} + 16\,x^{4} - 9\,n x - 42\,n x^{2} - 36\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}'(x) + \left(9\,x^{2} + 39\,x^{3} + 24\,x^{4} + 4\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2918633
327907216205
481378522264441289
52431458291624489001208085
67295346140941706410224295232850737
7218718954622089979285320396009408896318365
86561656102580665190485861163849121455842937624481997721
9196832230741023516248443235196123038472161481651494490288668812535525
1059049747954392347811179944191581202070943214403096641779217681762740801827278659393
1117714724800581464415247974032967878721265310721098982086382540824459408593020882368049920493581165
12531441814876253498394198247176461238300710477568745329600539110080268341984901540801951228824549121363843097180969
1315943232657205019202734879456334420973417483730584600460970611240204408322485994976108155347232334508863191808726585637260819434554165
1447829698609344267918159831050326169177631776185503614482647562954427173667456202241862721090492243242107506561133647488201935232213776641017984121950218577
Technical details
Node id
n-3a065c529988931c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ea8a87f06e938817

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,      2;
  2:     9,     18,       6;
  3:    27,     90,      72,      16;
  4:    81,    378,     522,     264,      44;
  5:   243,   1458,    2916,    2448,     900,     120;
  6:   729,   5346,   14094,   17064,   10224,    2952,     328;
  7:  2187,  18954,   62208,   99792,   85320,   39600,    9408,    896;
  8:  6561,  65610,  258066,  519048,  586116,  384912,  145584,  29376,  2448;
  9: 19683, 223074, 1023516, 2484432, 3519612, 3038472, 1614816, 514944, 90288, 6688;
%F Formula
The row polynomials satisfy 3*n*P(n,x) = (9*n+(12*n^2-6*n)*x+(12*n^2-30*n+18)*x^2+(4*n^2-12*n+8)*x^3)*P(n-1,x) - (9*n*x+(42*n-57)*x^2+(36*n-66)*x^3+(8*n-16)*x^4)*d/dx P(n-1,x) + (9*x^2+39*x^3+24*x^4+4*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[((9*n + (12*n^2 - 6*n)*x + (12*n^2 - 30*n + 18)*x^2 + (4*n^2 - 12*n + 8)*x^3)*P[n - 1, x] - (9*n*x + (42*n - 57)*x^2 + (36*n - 66)*x^3 + (8*n - 16)*x^4)*D[P[n - 1, x], x] + (9*x^2 + 39*x^3 + 24*x^4 + 4*x^5)*D[P[n - 1, x], {x, 2}])/(3*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] = ((9*n + (12*n^2 - 6*n)*x + (12*n^2 - 30*n + 18)*x^2 + (4*n^2 - 12*n + 8)*x^3)*v[i - 1] - (9*n*x + (42*n - 57)*x^2 + (36*n - 66)*x^3 + (8*n - 16)*x^4)*deriv(v[i - 1], x) + (9*x^2 + 39*x^3 + 24*x^4 + 4*x^5)*deriv(deriv(v[i - 1], x), x))/(3*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

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