Real-rooted atlas

A113413 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(1 + 3\,n + 2\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(4 + 11\,x + 9\,x^{2} + 2\,x^{3} - 8\,n - 16\,n x - 9\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(8 + 24\,x + 25\,x^{2} + 10\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
278217
3172814261
4418866202217
599262262122262773
62397529405921963222753
757721043156250811322883829805
8139357761010897045560193639844234921
9336315626312523518024464108523058526502124373
10811941784940141214249937253664193024552672562442961
11196011106762766904032943798122420161061783199064728366221577629
124732129085679985412987161383714101688052733819401250158887210186825618809
131142437593102277930407727048492504039302241975010535303327907521011806121874220011685
1427580719710406405880125310401645820015328448104230205248160196328054224010871215328143680271272673
Technical details
Node id
n-195811b74d9080d1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5db7e795cb165ce2

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:    7,     8,     2;
  3:   17,    28,    14,     2;
  4:   41,    88,    66,    20,     2;
  5:   99,   262,   262,   122,    26,     2;
  6:  239,   752,   940,   592,   196,    32,    2;
  7:  577,  2104,  3156,  2508,  1132,   288,   38,   2;
  8: 1393,  5776, 10108,  9704,  5560,  1936,  398,  44,  2;
  9: 3363, 15626, 31252, 35180, 24464, 10852, 3058, 526, 50, 2;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (2*n^2+3*n+1+(2*n^2+2*n)*x)*P(n-1,x) - (8*n-4+(16*n-11)*x+(9*n-9)*x^2+(n-2)*x^3)*d/dx P(n-1,x) + (8+24*x+25*x^2+10*x^3+x^4)*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 + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x] - (8*n - 4 + (16*n - 11)*x + (9*n - 9)*x^2 + (n - 2)*x^3)*D[P[n - 1, x], x] + (8 + 24*x + 25*x^2 + 10*x^3 + x^4)*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 + 1 + (2*n^2 + 2*n)*x)*v[i - 1] - (8*n - 4 + (16*n - 11)*x + (9*n - 9)*x^2 + (n - 2)*x^3)*deriv(v[i - 1], x) + (8 + 24*x + 25*x^2 + 10*x^3 + x^4)*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