Real-rooted atlas

A008288 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(144 - 72\,x + 864\,x^{2} + 243\,x^{3} - 243\,x^{4} - 252\,n + 342\,n x - 1269\,n x^{2} - 486\,n x^{3} + 243\,n x^{4} + 108\,n^{2} - 432\,n^{2} x + 405\,n^{2} x^{2} + 243\,n^{2} x^{3} + 162\,n^{3} x\right)\,P_{n}(x) = \left(480 - 648\,x - 5724\,x^{2} - 1701\,x^{3} + 1944\,x^{4} + 729\,x^{5} - 1860\,n - 3870\,n x + 3159\,n x^{2} + 2835\,n x^{3} + 1332\,n^{2} + 3348\,n^{2} x - 81\,n^{2} x^{2} - 729\,n^{2} x^{3} - 216\,n^{3} - 486\,n^{3} x\right)\,P_{n-1}(x) + \left(64 + 48\,x - 1152\,x^{2} - 1674\,x^{3} + 2349\,x^{4} + 2916\,x^{5} + 729\,x^{6} - 240\,n - 1080\,n x - 54\,n x^{2} + 4131\,n x^{3} + 3402\,n x^{4} + 729\,n x^{5} + 144\,n^{2} + 864\,n^{2} x + 1458\,n^{2} x^{2} + 486\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2538
32033962
46619815327444
52038941251621813050
6595347772276453234924320344
71690123393432645603296468505729132838
8469241148143982256986248184126360298892187853428

All 15 rows

Technical details
Node id
n-6870d0dead662dd7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5a8f3aa50606b336

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     5,      3;
   3:    20,     33,       9;
   4:    66,    198,     153,      27;
   5:   203,    894,    1251,     621,      81;
   6:   595,   3477,    7227,    6453,    2349,     243;
   7:  1690,  12339,   34326,   45603,   29646,    8505,     729;
   8:  4692,  41148,  143982,  256986,  248184,  126360,   29889,    2187;
   9: 12805, 131196,  554274, 1248966, 1622106, 1223748,  511029,  102789,   6561;
  10: 34481, 404343, 2003922, 5469822, 8984682, 9112014, 5622777, 1987983, 347733, 19683;
%F Formula
The row polynomials satisfy (108*n^2-252*n+144+(162*n^3-432*n^2+342*n-72)*x+(405*n^2-1269*n+864)*x^2+(243*n^2-486*n+243)*x^3+(243*n-243)*x^4)*P(n,x) = -(216*n^3-1332*n^2+1860*n-480+(486*n^3-3348*n^2+3870*n+648)*x+(81*n^2-3159*n+5724)*x^2+(729*n^2-2835*n+1701)*x^3-1944*x^4-729*x^5)*P(n-1,x) + (144*n^2-240*n+64+(864*n^2-1080*n+48)*x+(1458*n^2-54*n-1152)*x^2+(486*n^2+4131*n-1674)*x^3+(3402*n+2349)*x^4+(729*n+2916)*x^5+729*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(216*n^3 - 1332*n^2 + 1860*n - 480 + (486*n^3 - 3348*n^2 + 3870*n + 648)*x + (81*n^2 - 3159*n + 5724)*x^2 + (729*n^2 - 2835*n + 1701)*x^3 - 1944*x^4 - 729*x^5)*P[n - 1, x] + (144*n^2 - 240*n + 64 + (864*n^2 - 1080*n + 48)*x + (1458*n^2 - 54*n - 1152)*x^2 + (486*n^2 + 4131*n - 1674)*x^3 + (3402*n + 2349)*x^4 + (729*n + 2916)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(108*n^2 - 252*n + 144 + (162*n^3 - 432*n^2 + 342*n - 72)*x + (405*n^2 - 1269*n + 864)*x^2 + (243*n^2 - 486*n + 243)*x^3 + (243*n - 243)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(216*n^3 - 1332*n^2 + 1860*n - 480 + (486*n^3 - 3348*n^2 + 3870*n + 648)*x + (81*n^2 - 3159*n + 5724)*x^2 + (729*n^2 - 2835*n + 1701)*x^3 - 1944*x^4 - 729*x^5)*v[i - 1] + (144*n^2 - 240*n + 64 + (864*n^2 - 1080*n + 48)*x + (1458*n^2 - 54*n - 1152)*x^2 + (486*n^2 + 4131*n - 1674)*x^3 + (3402*n + 2349)*x^4 + (729*n + 2916)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(108*n^2 - 252*n + 144 + (162*n^3 - 432*n^2 + 342*n - 72)*x + (405*n^2 - 1269*n + 864)*x^2 + (243*n^2 - 486*n + 243)*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, v[i])))};
T(10)

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.