Real-rooted atlas

A144438 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
217715
3117453093
4137194314157703
5177711209424309726285
611572402114512312820883696164983
7131777555620017570026859019831056660763533
8163724374258379117335327003193312311206760451690110053879
9112777535111390847222615234739324253270043506220235155155225670146692365
1012557230522488275142091248185611014465920096694323990608575530289996355579992712349633335
111511770051520529720236084430137457527145914143179347959320118121346009052740585385669307170121692240994053869
121102372119454851540191288184023971093220541989629378112166520206191601089366209951651370142902342101550429155429173819257773914367159
1312047763941913497995246887920445662513115183635145907821239215186260274784785978540366875012803902258803366238825252537283937546238012913468652015719782061901
14140957192542421426904551362758733184402087298203019360432328128735387779193603772356398968149121220222877299620918427583898993832742157109098419136233588193991946194117057341853288862391
Technical details
Node id
n-f63b8f7fcf81ceba · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9c4ccaa67290bb6b

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: 1,    2;
  2: 1,    7,     7;
  3: 1,   17,    45,      30;
  4: 1,   37,   194,     314,     157;
  5: 1,   77,   711,    2094,    2430,      972;
  6: 1,  157,  2402,   11451,   23128,    20883,     6961;
  7: 1,  317,  7755,   56200,  175700,   268590,   198310,    56660;
  8: 1,  637, 24374,  258379, 1173353,  2700319,  3312311,  2067604,   516901;
  9: 1, 1277, 75351, 1139084, 7222615, 23473932, 42532700, 43506220, 23515515, 5225670;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (x+3*x^2+3*x^3+x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((x + 3*x^2 + 3*x^3 + x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((x + 3*x^2 + 3*x^3 + x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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