Real-rooted atlas

A257180 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-36 - 18\,x - 6\,x^{2} - 12\,n - 24\,n x - 11\,n x^{2} - 6\,n^{2} x - 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-9\,x + 3\,x^{2} + 6\,x^{3} - 3\,n x - 2\,n x^{2} + 5\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(17 + 8\,x + 7\,n + 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 24 + 18\,x\)
\(P_{2}(x) = 141 + 282\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
1241842
214128281504
3726282021783246048
43471228123421813884121575600
515828162654410616325308791404374997920
66987310665184171563556208426662954192381530913970880
7301362659030437827594778691766304599019770912210953452488207567360
81277619389787603159346689485785681185723210631869336136425660102209521771473269185920
95346672223001418248028940810431265992189715713601564689898857873419528920056724812004859049054486432000
10221543251243327218185612598451060893873842691243132903229491759481856564279224949669292055949724812215432501948617958961203200
1191097358679264854013374744899410148575227483479111848620581308516706448707565880682029715045496401242346982339632427001002070938637729217784371404800
12372361047365139687249349413404949243961843140417871857267459406710868701610974496013485266859497162792207989141470653116471134980200826827982446833256420726199346795242393600
13151495335619374067503063764413861688091338280387647295691346106013981948297761782194899202697040186425977303490485132244422990820228345700969023380285082616116244405018019694393628669615667095964190515200
146141123897101718188254242617813505183685420585502268509849777825146519394333021824866398924923424313514559141981992154429091499532737299101969955565029301884906610131237170471254020732661168223652385975734558215233605152056375511040000
152482277591452947076252082801192462121065650300368619536527828424723014102540855625371466486704455857298616341003011236038273968862001467366964958423475937561911080116122253439063102169509011058586081213850066919126370629533764563723416387106887475363406062811447296000
Technical details
Node id
n-7b43ff23f0ec92e7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9c764471e214691a

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       24,         18;
   2:      141,        282,          81;
   3:      726,       2820,        2178,          324;
   4:     3471,      22812,       34218,        13884,         1215;
   5:    15828,     162654,      410616,       325308,        79140,         4374;
   6:    69873,    1066518,     4171563,      5562084,      2666295,       419238,        15309;
   7:   301362,    6590304,    37827594,     77869176,     63045990,     19770912,      2109534,       52488;
   8:  1277619,   38978760,   315934668,    948578568,   1185723210,    631869336,    136425660,    10220952,     177147;
   9:  5346672,  223001418,  2480289408,  10431265992,  18971571360,  15646898988,   5787341952,   892005672,   48120048,    590490;
  10: 22154325, 1243327218, 18561259845, 106089387384, 269124313290, 322949175948, 185656427922, 49496692920, 5594972481, 221543250, 1948617;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(12*n+36+(6*n^2+24*n+18)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((3*n+9)*x+(n^2+2*n-3)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (7*n+17+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 24 + 18*x, P(2,x) = 141 + 282*x + 81*x^2.
%t Mathematica
P[1, x_] := 24 + 18*x;
P[2, x_] := 141 + 282*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 36 + (6*n^2 + 24*n + 18)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((3*n + 9)*x + (n^2 + 2*n - 3)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (7*n + 17 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 24 + 18*x; v[2] = 141 + 282*x + 81*x^2; for(n = 3, N, my(i = n); v[i] = (-(12*n + 36 + (6*n^2 + 24*n + 18)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((3*n + 9)*x + (n^2 + 2*n - 3)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (7*n + 17 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 1], x))/(n + 2)); 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.