Real-rooted atlas

A081579 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-90\,n - 90\,n x - 1269\,n x^{2} + 972\,n x^{3} + 243\,n x^{4} + 270\,n^{2} + 135\,n^{2} x + 1377\,n^{2} x^{2} + 243\,n^{2} x^{3} + 405\,n^{3} x\right)\,P_{n}(x) = \left(-1920 - 10620\,x - 11232\,x^{2} + 12069\,x^{3} + 6318\,x^{4} + 729\,x^{5} + 300\,n + 9090\,n x + 22680\,n x^{2} - 3483\,n x^{3} - 1458\,n x^{4} + 3870\,n^{2} + 7965\,n^{2} x - 9558\,n^{2} x^{2} - 2187\,n^{2} x^{3} - 1350\,n^{3} - 3645\,n^{3} x\right)\,P_{n-1}(x) + \left(-200 - 780\,x + 3870\,x^{2} + 23679\,x^{3} + 34425\,x^{4} + 9477\,x^{5} + 729\,x^{6} + 300\,n + 4860\,n x + 21735\,n x^{2} + 32562\,n x^{3} + 9234\,n x^{4} + 729\,n x^{5} + 900\,n^{2} + 5400\,n^{2} x + 8505\,n^{2} x^{2} + 1215\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
23242983
313236020727726
456022802520864816305
52176120002124013824332124352804
68320578881418401477446609612150729434767
73091226035282238412126248615162896564301121873522642
8113152111936043384328358336845769644692561195560148716656128207069
940832046425602136384051107328682305125130216021345120472392050519719683223648640
101458176187392009992448028635724847868019247694873628121904095878080180558721692738590491759012811
115160960739860484488652801501452288302868979237996646402971077408142802352041079733267237128560965517714713740741198
121813299228691251219521699847473282048177030904322694533299226608476672169372477446831523152169541488824517144818423288531441106715709593
136331187210960281608267692032356697907209719890099217458544563220993147136016893353779290014190624311577166086787800648878649120600528331594323824646182716
1421987328041340518403424720896016448082739250731635916810527092812801499293359360147231219456099145800921645200293142413666772952026501191200310361544019450740647829696344645923015
Technical details
Node id
n-e43dfe32a805d7dd · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-88cb068a99a74ff5

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:      5,       3;
  2:     32,      42,        9;
  3:    132,     360,      207,       27;
  4:    560,    2280,     2520,      864,       81;
  5:   2176,   12000,    21240,    13824,     3321,      243;
  6:   8320,   57888,   141840,   147744,    66096,    12150,      729;
  7:  30912,  260352,   822384,  1212624,   861516,   289656,    43011,    2187;
  8: 113152, 1119360,  4338432,  8358336,  8457696,  4469256,  1195560,  148716,   6561;
  9: 408320, 4642560, 21363840, 51107328, 68230512, 51302160, 21345120, 4723920, 505197, 19683;
%F Formula
The row polynomials satisfy (270*n^2-90*n+(405*n^3+135*n^2-90*n)*x+(1377*n^2-1269*n)*x^2+(243*n^2+972*n)*x^3+243*n*x^4)*P(n,x) = -(1350*n^3-3870*n^2-300*n+1920+(3645*n^3-7965*n^2-9090*n+10620)*x+(9558*n^2-22680*n+11232)*x^2+(2187*n^2+3483*n-12069)*x^3+(1458*n-6318)*x^4-729*x^5)*P(n-1,x) + (900*n^2+300*n-200+(5400*n^2+4860*n-780)*x+(8505*n^2+21735*n+3870)*x^2+(1215*n^2+32562*n+23679)*x^3+(9234*n+34425)*x^4+(729*n+9477)*x^5+729*x^6)*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[Together[(-(1350*n^3 - 3870*n^2 - 300*n + 1920 + (3645*n^3 - 7965*n^2 - 9090*n + 10620)*x + (9558*n^2 - 22680*n + 11232)*x^2 + (2187*n^2 + 3483*n - 12069)*x^3 + (1458*n - 6318)*x^4 - 729*x^5)*P[n - 1, x] + (900*n^2 + 300*n - 200 + (5400*n^2 + 4860*n - 780)*x + (8505*n^2 + 21735*n + 3870)*x^2 + (1215*n^2 + 32562*n + 23679)*x^3 + (9234*n + 34425)*x^4 + (729*n + 9477)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(270*n^2 - 90*n + (405*n^3 + 135*n^2 - 90*n)*x + (1377*n^2 - 1269*n)*x^2 + (243*n^2 + 972*n)*x^3 + 243*n*x^4)]];
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] = (-(1350*n^3 - 3870*n^2 - 300*n + 1920 + (3645*n^3 - 7965*n^2 - 9090*n + 10620)*x + (9558*n^2 - 22680*n + 11232)*x^2 + (2187*n^2 + 3483*n - 12069)*x^3 + (1458*n - 6318)*x^4 - 729*x^5)*v[i - 1] + (900*n^2 + 300*n - 200 + (5400*n^2 + 4860*n - 780)*x + (8505*n^2 + 21735*n + 3870)*x^2 + (1215*n^2 + 32562*n + 23679)*x^3 + (9234*n + 34425)*x^4 + (729*n + 9477)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(270*n^2 - 90*n + (405*n^3 + 135*n^2 - 90*n)*x + (1377*n^2 - 1269*n)*x^2 + (243*n^2 + 972*n)*x^3 + 243*n*x^4)); 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

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.