Real-rooted atlas

A063967 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 4\,x - 6\,x^{2} - x^{3} - 13\,n - 3\,n x + 5\,n x^{2} + n x^{3} + 6\,n^{2} + 7\,n^{2} x + n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(48\,x + 50\,x^{2} + 13\,x^{3} + x^{4} + 26\,n + 31\,n x - n x^{3} - 11\,n^{2} x - 2\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(96 + 244\,x + 224\,x^{2} + 91\,x^{3} + 16\,x^{4} + x^{5} + 84\,n + 140\,n x + 75\,n x^{2} + 15\,n x^{3} + n x^{4} + 12\,n^{2} + 8\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3117119
4363611184
511615876151366
63646324191311911566
71124237620178732012316615
83424854488644928157428627127648
910320297123648024844102112576386311114561
103083210065614287211536058167189013933501351471258
11914403339045382485031523011631201673222256996313911926666
1226950410887041965424208897614490926869482263095158779287764317835292
13790080349891269948808334176657910836172421416570396837785981067493647131718061
142305472111080962436539232174320284934121783944881068352700583657422115046139911111511127881180
156700096348961288333777612082728011866815683397480431776851669670548343301039810162911179331301551513757647
Technical details
Node id
n-f0df2fc32c70d89e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d69c8fc4fbc93bd8

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,      1;
   3:    11,      7,      1;
   4:    36,     36,     11,      1;
   5:   116,    158,     76,     15,     1;
   6:   364,    632,    419,    131,    19,     1;
   7:  1124,   2376,   2017,    873,   201,    23,    1;
   8:  3424,   8544,   8864,   4928,  1574,   286,   27,   1;
   9: 10320,  29712,  36480,  24844, 10211,  2576,  386,  31,  1;
  10: 30832, 100656, 142872, 115360, 58167, 18901, 3933, 501, 35, 1;
%F Formula
The row polynomials satisfy (n^3+6*n^2-13*n+6+(7*n^2-3*n-4)*x+(n^2+5*n-6)*x^2+(n-1)*x^3)*P(n,x) = -(2*n^3-26*n+(11*n^2-31*n-48)*x+(2*n^2-50)*x^2+(n-13)*x^3-x^4)*P(n-1,x) + (12*n^2+84*n+96+(8*n^2+140*n+244)*x+(n^2+75*n+224)*x^2+(15*n+91)*x^3+(n+16)*x^4+x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 26*n + (11*n^2 - 31*n - 48)*x + (2*n^2 - 50)*x^2 + (n - 13)*x^3 - x^4)*P[n - 1, x] + (12*n^2 + 84*n + 96 + (8*n^2 + 140*n + 244)*x + (n^2 + 75*n + 224)*x^2 + (15*n + 91)*x^3 + (n + 16)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 + 6*n^2 - 13*n + 6 + (7*n^2 - 3*n - 4)*x + (n^2 + 5*n - 6)*x^2 + (n - 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 26*n + (11*n^2 - 31*n - 48)*x + (2*n^2 - 50)*x^2 + (n - 13)*x^3 - x^4)*v[i - 1] + (12*n^2 + 84*n + 96 + (8*n^2 + 140*n + 244)*x + (n^2 + 75*n + 224)*x^2 + (15*n + 91)*x^3 + (n + 16)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 + 6*n^2 - 13*n + 6 + (7*n^2 - 3*n - 4)*x + (n^2 + 5*n - 6)*x^2 + (n - 1)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.