Real-rooted atlas

A081578 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(24\,x^{2} + 32\,x^{3} + n - 3\,n x - 40\,n x^{2} - 48\,n x^{3} + 5\,n^{2} x + 16\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(25\,x^{2} + 88\,x^{3} + 64\,x^{4} - n x - 18\,n x^{2} - 48\,n x^{3} - 32\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 17\,x^{3} + 32\,x^{4} + 16\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
217715
3112302063
41177812261279
51221514044551821215
6127249991186316415475319
71323722008551079445740164023247
81375203580132462752632140196844921101655
9142693583227546765721275541250166642914762444447
10147891888951510182142403014559122471933221435442871943271
111521114128768886338529610796521977528234788717396367307301328608496495
1215713621791814395574421925512005892300918935195285756290106239148639858137149111
1316216352414022176113373465462115153895443014549140883522376098252238279677723231195742162426303
14167193331667327881226648710801329362000318625507914641945717553064714505848978582691251105893587227710173575
Technical details
Node id
n-f99f79888b04bfd9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-0e2849de7a670bd3

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,  7,   7;
  3: 1, 12,  30,   20;
  4: 1, 17,  78,  122,    61;
  5: 1, 22, 151,  404,   455,   182;
  6: 1, 27, 249,  991,  1863,  1641,    547;
  7: 1, 32, 372, 2008,  5510,  7944,   5740,   1640;
  8: 1, 37, 520, 3580, 13246, 27526,  32140,  19684,  4921;
  9: 1, 42, 693, 5832, 27546, 76572, 127554, 125016, 66429, 14762;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+(5*n^2-3*n)*x+(16*n^2-40*n+24)*x^2+(16*n^2-48*n+32)*x^3)*P(n-1,x) - (n*x+(18*n-25)*x^2+(48*n-88)*x^3+(32*n-64)*x^4)*d/dx P(n-1,x) + (x^2+17*x^3+32*x^4+16*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 + (5*n^2 - 3*n)*x + (16*n^2 - 40*n + 24)*x^2 + (16*n^2 - 48*n + 32)*x^3)*P[n - 1, x] - (n*x + (18*n - 25)*x^2 + (48*n - 88)*x^3 + (32*n - 64)*x^4)*D[P[n - 1, x], x] + (x^2 + 17*x^3 + 32*x^4 + 16*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 + (5*n^2 - 3*n)*x + (16*n^2 - 40*n + 24)*x^2 + (16*n^2 - 48*n + 32)*x^3)*v[i - 1] - (n*x + (18*n - 25)*x^2 + (48*n - 88)*x^3 + (32*n - 64)*x^4)*deriv(v[i - 1], x) + (x^2 + 17*x^3 + 32*x^4 + 16*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