Real-rooted atlas

A075501 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-72 + 24\,x + 5\,x^{2}\right)\,P_{n-2}(x) + \left(-36\,x + 30\,x^{2}\right)\,P_{n-2}'(x) + \left(18 - 4\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 54 + 20\,x + x^{2}\)
\(P_{1}(x) = 468 + 324\,x + 39\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
05420175
1468324391832
245365040108064110721
34795279056264602700951156264
4536544127008061689695760567013212525083
562052482094854414125104311169627468010584175144676032
6730632963536524803223929609645350411866176671328181442241858118113
786836147260818895367385411520291390048047859336037204272145908029160279117766849160
810369949184106050954240170272751616868464115201848988944018932538241008897122898720445503401394027043147
912413705932818677397104643953243314944257166173721669492525696091028095488632132424024459019253638206534040719309306518400
101487830726656331245805363209238566564864075967661113344256582523957764209323728896370058502528184986996485423623929369360926644801233222423176705
1117843084808192590302010658816217128323962368022438051617945609366077552832001894029950292482063883773952012951313689604871087020811182691521560358812776455916171228061177448
1221405171422822410554129156096000512699197952655366634369268287488033942364252392960835878981114777611110219998958088569750565376040181297270401176388012802170376208249631201719906441171883804881595227
1325682287499182081891185097736355841215181814727389184196474763046109593612245474573705318403638134417479045125825520339230592054354776501076483111255551289601128996677952026431887081640025119683859128022680073515023990447754998912
14308163940740956163393862611850690560288871051852092211205828776666335520358444059982574025138176156778592577670103042994694128624586752333784290639900672229541756588052481011064198510080291958584952325586486796807065309888579196802937608321153689866107033861185
Technical details
Node id
n-65e2e6cbaaf8da19 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3928142656cba953

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           54,            20,             1;
  1:          468,           324,            39,             1;
  2:         4536,          5040,          1080,            64,            1;
  3:        47952,         79056,         26460,          2700,           95,           1;
  4:       536544,       1270080,        616896,         95760,         5670,         132,          1;
  5:      6205248,      20948544,      14125104,       3111696,       274680,       10584,        175,         1;
  6:     73063296,     353652480,     322392960,      96453504,     11866176,      671328,      18144,       224,       1;
  7:    868361472,    6081889536,    7385411520,    2913900480,    478593360,    37204272,    1459080,     29160,     279,     1;
  8:  10369949184,  106050954240,  170272751616,   86846411520,  18489889440,  1893253824,  100889712,   2898720,   44550,   340,   1;
  9: 124137059328, 1867739710464, 3953243314944, 2571661737216, 694925256960, 91028095488, 6321324240, 244590192, 5363820, 65340, 407, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(72-24*x-5*x^2)*P(n-2,x) - (36*x-30*x^2)*d/dx P(n-2,x) + (18-4*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 54 + 20*x + x^2, P(1,x) = 468 + 324*x + 39*x^2 + x^3.
%t Mathematica
P[0, x_] := 54 + 20*x + x^2;
P[1, x_] := 468 + 324*x + 39*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(72 - 24*x - 5*x^2)*P[n - 2, x] - (36*x - 30*x^2)*D[P[n - 2, x], x] + (18 - 4*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 54 + 20*x + x^2; v[2] = 468 + 324*x + 39*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(72 - 24*x - 5*x^2)*v[i - 2] - (36*x - 30*x^2)*deriv(v[i - 2], x) + (18 - 4*x)*v[i - 1] + 6*x*deriv(v[i - 1], x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.