Real-rooted atlas

A144698 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
21613130
364113321210
42568215317111680
51024538564702090150115120
6409633069657454051074703091151200
7163841940175928846122952218002519162811663200
86553611011574915423784336948358751113919818531267119958400
9262144609560938301794893115448574721434204646527420625949625461259459200
10104857633099485284671245931595160131115852082232509822376529023951280809040510513632428800
11419430417703176120390223209083207325179420294401657903552271716034881382290530105513600249348510224154486432000
121677721693549045314181483291839726744752252851014452927475552181877692835345823481675881776158504544697057623591204631871782912000
136710886448955560739631478518274380503752626420075563104662293909967421967327708419549679690764475402907484679618541519246090942317771840942114820309504000
1426843545625417304461641530940041636241268486629317915863266683807207528038415737796054355262033876508190552592725083293592154883260925454391804610214670188226819011266765571072000
Technical details
Node id
n-efec96f0fcf87129 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-197c535f804b8fcf

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:      4,       1;
  2:     16,      13,        1;
  3:     64,     113,       32,        1;
  4:    256,     821,      531,       71,        1;
  5:   1024,    5385,     6470,     2090,      150,        1;
  6:   4096,   33069,    65745,    40510,     7470,      309,       1;
  7:  16384,  194017,   592884,   612295,   221800,    25191,     628,      1;
  8:  65536, 1101157,  4915423,  7843369,  4835875,  1113919,   81853,   1267,    1;
  9: 262144, 6095609, 38301794, 89311544, 85747214, 34204646, 5274206, 259496, 2546, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+n*x)*P(n-1,x) + (x-x^2)*d/dx 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[(4 + n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
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] = (4 + n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], 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