Real-rooted atlas

A008292 → γ (scale 1/4) (shift +1) → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,x + n\right)\,P_{n}(x) = \left(-4\,x^{2} - 12\,x^{3} + 16\,x^{4} + 2\,n + 14\,n x + 22\,n x^{2} + 10\,n x^{3} - 8\,n x^{4} + n^{2} - 7\,n^{2} x^{2} - 6\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x^{3} + 20\,x^{4} + 16\,x^{5} + 2\,n x + 12\,n x^{2} + 22\,n x^{3} + 12\,n x^{4} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4\,x - 18\,x^{2} + 8\,x^{3} + 4\,n x + 10\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x^{2} - 8\,x^{3} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
28210
3401656
424015216408
5168015042723456
61344015792451227234016
7120960177408721927936378496
812096002136960115200020070479364707200
9133056002757888018720000467148835379264629760
1015966720038054016031350528010435814412599296353792971023872
1120756736005598996480544693248022984642563993098242236825615841744896
122905943040087593425920985253068805076943872011827845120106371993622368256278861535232
134358914560001452971520000185841976320011369156198403375674265604386175795219037573125267531300864
146974263296000254851204608003657336883200025990477793280947087511552016572263362561163552153601903757312106269590806528
Technical details
Node id
n-8424272eedafea66 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e639584d673dfbeb

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:        8,        2;
  3:       40,       16;
  4:      240,      152,       16;
  5:     1680,     1504,      272;
  6:    13440,    15792,     4512,     272;
  7:   120960,   177408,    72192,    7936;
  8:  1209600,  2136960,  1152000,  200704,   7936;
  9: 13305600, 27578880, 18720000, 4671488, 353792;
%F Formula
The row polynomials satisfy (n+4*x)*P(n,x) = -(n^3-n^2-2*n+(2*n^3-14*n)*x+(n^3+7*n^2-22*n+4)*x^2+(6*n^2-10*n+12)*x^3+(8*n-16)*x^4)*P(n-2,x) + ((2*n^2+2*n)*x+(4*n^2+12*n)*x^2+(2*n^2+22*n+4)*x^3+(12*n+20)*x^4+16*x^5)*d/dx P(n-2,x) + (2*n^2+(2*n^2+4*n-4)*x+(10*n-18)*x^2+8*x^3)*P(n-1,x) - (2*n*x+(2*n+8)*x^2+8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 2*n + (2*n^3 - 14*n)*x + (n^3 + 7*n^2 - 22*n + 4)*x^2 + (6*n^2 - 10*n + 12)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] + ((2*n^2 + 2*n)*x + (4*n^2 + 12*n)*x^2 + (2*n^2 + 22*n + 4)*x^3 + (12*n + 20)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (2*n^2 + (2*n^2 + 4*n - 4)*x + (10*n - 18)*x^2 + 8*x^3)*P[n - 1, x] - (2*n*x + (2*n + 8)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(n + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - 2*n + (2*n^3 - 14*n)*x + (n^3 + 7*n^2 - 22*n + 4)*x^2 + (6*n^2 - 10*n + 12)*x^3 + (8*n - 16)*x^4)*v[i - 2] + ((2*n^2 + 2*n)*x + (4*n^2 + 12*n)*x^2 + (2*n^2 + 22*n + 4)*x^3 + (12*n + 20)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (2*n^2 + (2*n^2 + 4*n - 4)*x + (10*n - 18)*x^2 + 8*x^3)*v[i - 1] - (2*n*x + (2*n + 8)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(n + 4*x)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.