Real-rooted atlas

A113413 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(2 + 2\,x^{2} + 4\,x^{3} + 2\,n + 3\,n x - 6\,n x^{2} - 6\,n x^{3} + 3\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x + 5\,x^{2} + 18\,x^{3} + 8\,x^{4} - 2\,n x - 10\,n x^{2} - 12\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 9\,x^{3} + 8\,x^{4} + 2\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
2412723
38364617107
4169619215241497
532240656816466992309
664576200034243060136423910727
712813445664124001525610596387057749835
825630721523240704641606124834720107361393231521
95126912394241245442401922937282283841092482928233631075589
1010241536099072361472826112123686412308808059523332527881281194996919
112048337922432001006848266368047308805763136482416027252889919402098701960123214443
1240967372858572827136008163072168028162428787224873216179416808906912289449655399247321107848529
13819215974413885447118848240102405626598494306816114579328101041248639719202831198483086561451698114243501037445
141638434406432481281825996868271104179592192342914304483150848505275456390782912220323312879371842352318837805162758072327695367
Technical details
Node id
n-6d6fa12db802db1c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d7ad33eaae520bf7

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:   2,    3;
  2:   4,   12,     7;
  3:   8,   36,    46,     17;
  4:  16,   96,   192,    152,     41;
  5:  32,  240,   656,    816,    466,     99;
  6:  64,  576,  2000,   3424,   3060,   1364,    239;
  7: 128, 1344,  5664,  12400,  15256,  10596,   3870,    577;
  8: 256, 3072, 15232,  40704,  64160,  61248,  34720,  10736,  1393;
  9: 512, 6912, 39424, 124544, 240192, 293728, 228384, 109248, 29282, 3363;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (2*n+2+(3*n^2+3*n)*x+(4*n^2-6*n+2)*x^2+(2*n^2-6*n+4)*x^3)*P(n-1,x) - ((2*n+2)*x+(10*n-5)*x^2+(12*n-18)*x^3+(4*n-8)*x^4)*d/dx P(n-1,x) + (2*x^2+9*x^3+8*x^4+2*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[((2*n + 2 + (3*n^2 + 3*n)*x + (4*n^2 - 6*n + 2)*x^2 + (2*n^2 - 6*n + 4)*x^3)*P[n - 1, x] - ((2*n + 2)*x + (10*n - 5)*x^2 + (12*n - 18)*x^3 + (4*n - 8)*x^4)*D[P[n - 1, x], x] + (2*x^2 + 9*x^3 + 8*x^4 + 2*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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 2 + (3*n^2 + 3*n)*x + (4*n^2 - 6*n + 2)*x^2 + (2*n^2 - 6*n + 4)*x^3)*v[i - 1] - ((2*n + 2)*x + (10*n - 5)*x^2 + (12*n - 18)*x^3 + (4*n - 8)*x^4)*deriv(v[i - 1], x) + (2*x^2 + 9*x^3 + 8*x^4 + 2*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

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