Real-rooted atlas

A257620 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + n\right)\,P_{n}(x) = \left(18 - 81\,x + 126\,x^{2} - 81\,x^{3} + 18\,x^{4} - 18\,n + 63\,n x - 108\,n x^{2} + 90\,n x^{3} - 18\,n x^{4} - 18\,n^{2} x + 45\,n^{2} x^{2} - 27\,n^{2} x^{3} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(9\,x - 45\,x^{2} + 81\,x^{3} - 63\,x^{4} + 18\,x^{5} - 9\,n x + 36\,n x^{2} - 54\,n x^{3} + 27\,n x^{4} - 9\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-15 + 36\,x - 27\,x^{2} + 6\,x^{3} + 9\,n - 21\,n x + 15\,n x^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 12\,x^{2} - 6\,x^{3} + 3\,n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
245549108
3324891378271620
42187127981166424308129160
51409416062329354412879015066243612360
68820918239586152031523422013056399185472914696640
75423761931777111186505017079376581102708126299255554982187396809280
8330018319497979818325529104677953634391927896011539618021188984423346110656111904278400
919958562190320831927820267128112126335264154750186692795489625981555098717611011457522011602619683392841187200
10120282813181463481903993543509492430217337256521753149434643786564816681485822233970202218260616100950760891208142545904914142282739200
1172346834817017183687554988777769224879719268940715569093835388820151994749629411102539492378826185861391238256723629022891957184847725239818177147551549026828800
124346655939157712025483073381413669516923994991169190422569416140707580548671096996126810162683808960259974229776850844273424205498532062212853110834194434644435250179053144123165059126809600
1326099067510144939856100319562265827720561671708167825797810648173458730627028804684566068104535676071504420572820840556440073970857377998674579474726349649535130539590426432052075471137466662611819938615943231042427660706432000
141566565836571324032479408701223530092190327529182333062458912425298377334491140099427722580891230786164733072620409028351405542213173933244059135700282551773471210996544993935722116562052474882857484987211811882068159771479035156718762254478296950036527713908736000
Technical details
Node id
n-08bb072bfddb6a12 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d762c6db5469cb8f

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:        6,          3;
  2:       45,         54,           9;
  3:      324,        891,         378,           27;
  4:     2187,      12798,       11664,         2430,           81;
  5:    14094,     160623,      293544,       128790,        15066,         243;
  6:    88209,    1823958,     6152031,      5234220,      1305639,       91854,         729;
  7:   542376,   19317771,   111865050,    170793765,     81102708,    12629925,      555498,       2187;
  8:  3300183,  194979798,  1832552910,   4677953634,   3919278960,  1153961802,   118898442,    3346110,     6561;
  9: 19958562, 1903208319, 27820267128, 112126335264, 154750186692, 79548962598, 15550987176, 1101145752, 20116026, 19683;
%F Formula
The row polynomials satisfy (n-2+2*x)*P(n,x) = -(18*n-18+(18*n^2-63*n+81)*x+(9*n^3-45*n^2+108*n-126)*x^2+(27*n^2-90*n+81)*x^3+(18*n-18)*x^4)*P(n-2,x) - ((9*n-9)*x+(9*n^2-36*n+45)*x^2-(9*n^2-54*n+81)*x^3-(27*n-63)*x^4-18*x^5)*d/dx P(n-2,x) + (9*n-15+(6*n^2-21*n+36)*x+(15*n-27)*x^2+6*x^3)*P(n-1,x) + ((3*n-6)*x-(3*n-12)*x^2-6*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n - 18 + (18*n^2 - 63*n + 81)*x + (9*n^3 - 45*n^2 + 108*n - 126)*x^2 + (27*n^2 - 90*n + 81)*x^3 + (18*n - 18)*x^4)*P[n - 2, x] - ((9*n - 9)*x + (9*n^2 - 36*n + 45)*x^2 - (9*n^2 - 54*n + 81)*x^3 - (27*n - 63)*x^4 - 18*x^5)*D[P[n - 2, x], x] + (9*n - 15 + (6*n^2 - 21*n + 36)*x + (15*n - 27)*x^2 + 6*x^3)*P[n - 1, x] + ((3*n - 6)*x - (3*n - 12)*x^2 - 6*x^3)*D[P[n - 1, x], x])/(n - 2 + 2*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] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n - 18 + (18*n^2 - 63*n + 81)*x + (9*n^3 - 45*n^2 + 108*n - 126)*x^2 + (27*n^2 - 90*n + 81)*x^3 + (18*n - 18)*x^4)*v[i - 2] - ((9*n - 9)*x + (9*n^2 - 36*n + 45)*x^2 - (9*n^2 - 54*n + 81)*x^3 - (27*n - 63)*x^4 - 18*x^5)*deriv(v[i - 2], x) + (9*n - 15 + (6*n^2 - 21*n + 36)*x + (15*n - 27)*x^2 + 6*x^3)*v[i - 1] + ((3*n - 6)*x - (3*n - 12)*x^2 - 6*x^3)*deriv(v[i - 1], x))/(n - 2 + 2*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.