Real-rooted atlas

A121574 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(33\,n + 369\,n x - 486\,n x^{2} + 81\,n x^{3} + 243\,n x^{4} + 261\,n^{2} + 486\,n^{2} x^{2} + 243\,n^{2} x^{3} - 81\,n^{3} + 243\,n^{3} x\right)\,P_{n}(x) = \left(-363 - 2151\,x - 2862\,x^{2} + 1782\,x^{3} + 2673\,x^{4} + 729\,x^{5} + 1092\,n + 4842\,n x + 5427\,n x^{2} + 1134\,n x^{3} - 243\,n x^{4} + 1197\,n^{2} + 2592\,n^{2} x - 729\,n^{2} x^{2} - 972\,n^{2} x^{3} - 648\,n^{3} - 972\,n^{3} x\right)\,P_{n-1}(x) + \left(-26 + 57\,x + 1773\,x^{2} + 6750\,x^{3} + 9396\,x^{4} + 4617\,x^{5} + 729\,x^{6} + 273\,n + 2268\,n x + 6777\,n x^{2} + 8667\,n x^{3} + 4374\,n x^{4} + 729\,n x^{5} + 351\,n^{2} + 1863\,n^{2} x + 2673\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
23139979
313830318027648
457418331899729815116
522869648146349747275424339312
68839465309387991557446319963729296128
7334502113415347086951694952341897833499221872196864

All 15 rows

Technical details
Node id
n-e084df69abd1b0bd · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8f36181a92323739

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:     31,      39,        9;
  3:    138,     303,      180,       27;
  4:    574,    1833,     1899,      729,       81;
  5:   2286,    9648,    14634,     9747,     2754,      243;
  6:   8839,   46530,    93879,    91557,    44631,     9963,      729;
  7:  33450,  211341,   534708,   695169,   495234,   189783,    34992,    2187;
  8: 124537,  918939,  2802663,  4581063,  4350996,  2430729,   766179,  120285,   6561;
  9: 457752, 3865389, 13817322, 27293247, 32623398, 24279102, 11128914, 2976507, 406782, 19683;
%F Formula
The row polynomials satisfy -(81*n^3-261*n^2-33*n-(243*n^3+369*n)*x-(486*n^2-486*n)*x^2-(243*n^2+81*n)*x^3-243*n*x^4)*P(n,x) = -(648*n^3-1197*n^2-1092*n+363+(972*n^3-2592*n^2-4842*n+2151)*x+(729*n^2-5427*n+2862)*x^2+(972*n^2-1134*n-1782)*x^3+(243*n-2673)*x^4-729*x^5)*P(n-1,x) + (351*n^2+273*n-26+(1863*n^2+2268*n+57)*x+(2673*n^2+6777*n+1773)*x^2+(729*n^2+8667*n+6750)*x^3+(4374*n+9396)*x^4+(729*n+4617)*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[(-(648*n^3 - 1197*n^2 - 1092*n + 363 + (972*n^3 - 2592*n^2 - 4842*n + 2151)*x + (729*n^2 - 5427*n + 2862)*x^2 + (972*n^2 - 1134*n - 1782)*x^3 + (243*n - 2673)*x^4 - 729*x^5)*P[n - 1, x] + (351*n^2 + 273*n - 26 + (1863*n^2 + 2268*n + 57)*x + (2673*n^2 + 6777*n + 1773)*x^2 + (729*n^2 + 8667*n + 6750)*x^3 + (4374*n + 9396)*x^4 + (729*n + 4617)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(-81*n^3 + 261*n^2 + 33*n + (243*n^3 + 369*n)*x + (486*n^2 - 486*n)*x^2 + (243*n^2 + 81*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] = (-(648*n^3 - 1197*n^2 - 1092*n + 363 + (972*n^3 - 2592*n^2 - 4842*n + 2151)*x + (729*n^2 - 5427*n + 2862)*x^2 + (972*n^2 - 1134*n - 1782)*x^3 + (243*n - 2673)*x^4 - 729*x^5)*v[i - 1] + (351*n^2 + 273*n - 26 + (1863*n^2 + 2268*n + 57)*x + (2673*n^2 + 6777*n + 1773)*x^2 + (729*n^2 + 8667*n + 6750)*x^3 + (4374*n + 9396)*x^4 + (729*n + 4617)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(-81*n^3 + 261*n^2 + 33*n + (243*n^3 + 369*n)*x + (486*n^2 - 486*n)*x^2 + (243*n^2 + 81*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.