Real-rooted atlas

A122542 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2}\right)\,P_{n-4}(x) + \left(-4 - 18\,x - 18\,x^{2}\right)\,P_{n-3}(x) + \left(-2 - 6\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 3\,x\)
\(P_{3}(x) = 15 + 30\,x + 9\,x^{2}\)
\(P_{4}(x) = 46 + 165\,x + 144\,x^{2} + 27\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
31530954
44616514427382
51376961098594812606
6392258359765832226824317294
7109588622709939150272978262729112494
8300228809109656211383218538117855291602187720590
98113900484102929919801373922109738848114010060265614560046
10216762731471449648422085673625767888266511174818851943411721968328573966
1157375809166490223716705764352241464762800041396994225129787158051114161459049177595374
1215069423515171601308862535915154823724253609866278428428202944852949376702651300137791361771471096265038
1339320967261685085660622395684663716957112281054761635400692150588509494320279938673158496249870124002905314416727609646
14102027218984111157801608773647632248786348455211522138662674924968094358276580516884199309001153149486434366518040389516159432341078592398
1526345195297451048013623925937031069304794243233807120464229395195555734684606535182672693704525620218047559213592414997412100911571307344864782969249724432494
Technical details
Node id
n-ba338b6b81ab1684 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0569c33d152ea32b

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:     4,      3;
   3:    15,     30,       9;
   4:    46,    165,     144,      27;
   5:   137,    696,    1098,     594,      81;
   6:   392,   2583,    5976,    5832,    2268,     243;
   7:  1095,   8862,   27099,   39150,   27297,    8262,     729;
   8:  3002,  28809,  109656,  211383,  218538,  117855,   29160,    2187;
   9:  8113,  90048,  410292,  991980, 1373922, 1097388,  481140,  100602,   6561;
  10: 21676, 273147, 1449648, 4220856, 7362576, 7888266, 5111748, 1885194, 341172, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(1+6*x+9*x^2)*P(n-4,x) - (4+18*x+18*x^2)*P(n-3,x) - (2+6*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 4 + 3*x, P(3,x) = 15 + 30*x + 9*x^2, P(4,x) = 46 + 165*x + 144*x^2 + 27*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 3*x;
P[3, x_] := 15 + 30*x + 9*x^2;
P[4, x_] := 46 + 165*x + 144*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 9*x^2)*P[n - 4, x] - (4 + 18*x + 18*x^2)*P[n - 3, x] - (2 + 6*x + 9*x^2)*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 3*x; v[3] = 15 + 30*x + 9*x^2; v[4] = 46 + 165*x + 144*x^2 + 27*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 6*x + 9*x^2)*v[i - 4] - (4 + 18*x + 18*x^2)*v[i - 3] - (2 + 6*x + 9*x^2)*v[i - 2] + (4 + 6*x)*v[i - 1]); 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

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.