Real-rooted atlas

A143685 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(60\,x^{2} + 80\,x^{3} + n - 9\,n x - 100\,n x^{2} - 120\,n x^{3} + 11\,n^{2} x + 40\,n^{2} x^{2} + 40\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(61\,x^{2} + 220\,x^{3} + 160\,x^{4} - n x - 42\,n x^{2} - 120\,n x^{3} - 80\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 41\,x^{3} + 80\,x^{4} + 40\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
21131327
31246644135
4135240410205891
5146535170020158065103
615795152451126510371345731347
716814881237641870663364958814168185895
8179214624424122386298666376984237796594491121931
919029254272029601010316521948170203508011088452464106711903
1011013825685956205802944962780094212012270107120255139305102786140330467
11111248461033801168575723469225639824544325527058581554858680235037664273412241805655
121123598814840620271151577441772199086202425336357769827400548965275883096106785438177975731451365371
13113472512050043297961312762421793773236426941521483271883224125392222059814811363813644481363051740558548706597903
1411458635274505509751557451853402143595179321032552402799851024955415513513358453118636651556653007145215802905530828986552244370387
Technical details
Node id
n-81faaf48151405fa · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ea9a5170d08e2562

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, 13,   13;
  3: 1, 24,   66,    44;
  4: 1, 35,  240,   410,    205;
  5: 1, 46,  535,  1700,   2015,     806;
  6: 1, 57,  951,  5245,  11265,   10371,    3457;
  7: 1, 68, 1488, 12376,  41870,   66336,   49588,   14168;
  8: 1, 79, 2146, 24424, 122386,  298666,  376984,  237796,   59449;
  9: 1, 90, 2925, 42720, 296010, 1031652, 1948170, 2035080, 1108845, 246410;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+(11*n^2-9*n)*x+(40*n^2-100*n+60)*x^2+(40*n^2-120*n+80)*x^3)*P(n-1,x) - (n*x+(42*n-61)*x^2+(120*n-220)*x^3+(80*n-160)*x^4)*d/dx P(n-1,x) + (x^2+41*x^3+80*x^4+40*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 + (11*n^2 - 9*n)*x + (40*n^2 - 100*n + 60)*x^2 + (40*n^2 - 120*n + 80)*x^3)*P[n - 1, x] - (n*x + (42*n - 61)*x^2 + (120*n - 220)*x^3 + (80*n - 160)*x^4)*D[P[n - 1, x], x] + (x^2 + 41*x^3 + 80*x^4 + 40*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 + (11*n^2 - 9*n)*x + (40*n^2 - 100*n + 60)*x^2 + (40*n^2 - 120*n + 80)*x^3)*v[i - 1] - (n*x + (42*n - 61)*x^2 + (120*n - 220)*x^3 + (80*n - 160)*x^4)*deriv(v[i - 1], x) + (x^2 + 41*x^3 + 80*x^4 + 40*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