Real-rooted atlas

A116412 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-9 - 1692\,x - 1404\,x^{2} + 162\,x^{3} + 243\,x^{4} - 741\,n + 396\,n x - 405\,n x^{2} + 81\,n x^{3} + 243\,n x^{4} + 639\,n^{2} - 81\,n^{2} x + 243\,n^{2} x^{2} + 243\,n^{2} x^{3} - 81\,n^{3} + 243\,n^{3} x\right)\,P_{n}(x) = \left(-2238 - 13626\,x - 21168\,x^{2} - 3888\,x^{3} + 4374\,x^{4} + 1458\,x^{5} - 1626\,n - 720\,n x + 5832\,n x^{2} + 3564\,n x^{3} + 486\,n x^{4} + 3168\,n^{2} + 6804\,n^{2} x + 972\,n^{2} x^{2} - 972\,n^{2} x^{3} - 648\,n^{3} - 972\,n^{3} x\right)\,P_{n-1}(x) + \left(-299 - 2562\,x - 6867\,x^{2} - 4590\,x^{3} + 4293\,x^{4} + 3888\,x^{5} + 729\,x^{6} - 312\,n - 1188\,n x + 459\,n x^{2} + 4779\,n x^{3} + 3645\,n x^{4} + 729\,n x^{5} + 351\,n^{2} + 1863\,n^{2} x + 2673\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
23742988
316934218927727
471221362079756815764
52860114811653310476283524344428
611125561781085131013044738510206729335440
7422892578716285877867265398651997463572121872492992

All 15 rows

Technical details
Node id
n-2d865788de1aa9aa · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4a06b02dc06909d3

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:      7,       3;
  2:     37,      42,        9;
  3:    169,     342,      189,       27;
  4:    712,    2136,     2079,      756,       81;
  5:   2860,   11481,    16533,    10476,     2835,      243;
  6:  11125,   56178,   108513,   101304,    47385,    10206,      729;
  7:  42289,  257871,   628587,   786726,   539865,   199746,    35721,    2187;
  8: 157987, 1130280,  3337371,  5276232,  4846230,  2620512,   801171,  122472,   6561;
  9: 582289, 4784328, 16619985, 31874310, 36974394, 26709831, 11895093, 3096792, 413343, 19683;
%F Formula
The row polynomials satisfy -(81*n^3-639*n^2+741*n+9-(243*n^3-81*n^2+396*n-1692)*x-(243*n^2-405*n-1404)*x^2-(243*n^2+81*n+162)*x^3-(243*n+243)*x^4)*P(n,x) = -(648*n^3-3168*n^2+1626*n+2238+(972*n^3-6804*n^2+720*n+13626)*x-(972*n^2+5832*n-21168)*x^2+(972*n^2-3564*n+3888)*x^3-(486*n+4374)*x^4-1458*x^5)*P(n-1,x) + (351*n^2-312*n-299+(1863*n^2-1188*n-2562)*x+(2673*n^2+459*n-6867)*x^2+(729*n^2+4779*n-4590)*x^3+(3645*n+4293)*x^4+(729*n+3888)*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 - 3168*n^2 + 1626*n + 2238 + (972*n^3 - 6804*n^2 + 720*n + 13626)*x - (972*n^2 + 5832*n - 21168)*x^2 + (972*n^2 - 3564*n + 3888)*x^3 - (486*n + 4374)*x^4 - 1458*x^5)*P[n - 1, x] + (351*n^2 - 312*n - 299 + (1863*n^2 - 1188*n - 2562)*x + (2673*n^2 + 459*n - 6867)*x^2 + (729*n^2 + 4779*n - 4590)*x^3 + (3645*n + 4293)*x^4 + (729*n + 3888)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(-81*n^3 + 639*n^2 - 741*n - 9 + (243*n^3 - 81*n^2 + 396*n - 1692)*x + (243*n^2 - 405*n - 1404)*x^2 + (243*n^2 + 81*n + 162)*x^3 + (243*n + 243)*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 - 3168*n^2 + 1626*n + 2238 + (972*n^3 - 6804*n^2 + 720*n + 13626)*x - (972*n^2 + 5832*n - 21168)*x^2 + (972*n^2 - 3564*n + 3888)*x^3 - (486*n + 4374)*x^4 - 1458*x^5)*v[i - 1] + (351*n^2 - 312*n - 299 + (1863*n^2 - 1188*n - 2562)*x + (2673*n^2 + 459*n - 6867)*x^2 + (729*n^2 + 4779*n - 4590)*x^3 + (3645*n + 4293)*x^4 + (729*n + 3888)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(-81*n^3 + 639*n^2 - 741*n - 9 + (243*n^3 - 81*n^2 + 396*n - 1692)*x + (243*n^2 - 405*n - 1404)*x^2 + (243*n^2 + 81*n + 162)*x^3 + (243*n + 243)*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.