Real-rooted atlas

A081578 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(9\,n\right)\,P_{n}(x) = \left(72\,x^{2} + 32\,x^{3} + 27\,n - 27\,n x - 120\,n x^{2} - 48\,n x^{3} + 45\,n^{2} x + 48\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(225\,x^{2} + 264\,x^{3} + 64\,x^{4} - 27\,n x - 162\,n x^{2} - 144\,n x^{3} - 32\,n x^{4}\right)\,P_{n-1}'(x) + \left(27\,x^{2} + 153\,x^{3} + 96\,x^{4} + 16\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2921737
3271089020245
481459702366611669
5243178240773636136518211285
67296561201692675716767492354776453
72187233289039616264814877071496172201640517685
865618091937908086994010729267432022892605905249213505861
9196832755621515591425152866936786202332344395811251441992871476223741525
105904992510158458511944024337550790442605063264413415096294424739766430544287160777957
1117714730705482192686284479436194343381280880784262355436160179768633929491565672421921901328601088788085
1253144110097379804247383526799949444887551627606953185982480014318289007443374312572715255661095471744583985817373275909
13159432332949342289635345142544286043649217638774327106119456455051121897757673253543133311565282101546527520144516423316969119574249931836565
14478296910681964110272754535609714049193610451694461126362170867519569791694677976287995259135579928051142179824073916579203707244219753317673587227338138493733
Technical details
Node id
n-3e7db72259e26388 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ecc774d95c805d20

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:     3,      2;
  2:     9,     21,       7;
  3:    27,    108,      90,      20;
  4:    81,    459,     702,     366,      61;
  5:   243,   1782,    4077,    3636,    1365,     182;
  6:   729,   6561,   20169,   26757,   16767,    4923,     547;
  7:  2187,  23328,   90396,  162648,  148770,   71496,   17220,    1640;
  8:  6561,  80919,  379080,  869940, 1072926,  743202,  289260,   59052,   4921;
  9: 19683, 275562, 1515591, 4251528, 6693678, 6202332, 3443958, 1125144, 199287, 14762;
%F Formula
The row polynomials satisfy 9*n*P(n,x) = (27*n+(45*n^2-27*n)*x+(48*n^2-120*n+72)*x^2+(16*n^2-48*n+32)*x^3)*P(n-1,x) - (27*n*x+(162*n-225)*x^2+(144*n-264)*x^3+(32*n-64)*x^4)*d/dx P(n-1,x) + (27*x^2+153*x^3+96*x^4+16*x^5)*d^2/dx^2 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[((27*n + (45*n^2 - 27*n)*x + (48*n^2 - 120*n + 72)*x^2 + (16*n^2 - 48*n + 32)*x^3)*P[n - 1, x] - (27*n*x + (162*n - 225)*x^2 + (144*n - 264)*x^3 + (32*n - 64)*x^4)*D[P[n - 1, x], x] + (27*x^2 + 153*x^3 + 96*x^4 + 16*x^5)*D[P[n - 1, x], {x, 2}])/(9*n)]];
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] = ((27*n + (45*n^2 - 27*n)*x + (48*n^2 - 120*n + 72)*x^2 + (16*n^2 - 48*n + 32)*x^3)*v[i - 1] - (27*n*x + (162*n - 225)*x^2 + (144*n - 264)*x^3 + (32*n - 64)*x^4)*deriv(v[i - 1], x) + (27*x^2 + 153*x^3 + 96*x^4 + 16*x^5)*deriv(deriv(v[i - 1], x), x))/(9*n)); 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