Real-rooted atlas

A038338 → basis A013610 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-1440 + 432\,x\right)\,P_{n-2}(x) + \left(84 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3600 + 5616\,x + 1296\,x^{2}\)
\(P_{1}(x) = 216000 + 575424\,x + 326592\,x^{2} + 46656\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
036005616129610512
1216000575424326592466561164672
2129600004957977648708864162362881679616129164544
37776000003895962624565470720031912704007457495046046617614325755904
4466560000002894583029765665277583364692981473281821241221123265173504021767823361588892848128
52799360000000207198504714245154910809292857663447810048324979090882569534306631680138443356569678364164096176226779824128
61679616000000001444579835314176438528571165900862684481668382724746186548379648200246560653312047049844123238457362568118272282110990745619545609587982336
710077696000000000987640626075402243550891336005058566236537814838149126024574244369203203393795218669568001137358670289960962224163051038310423358790033735681015599566684162167836556495159296

All 15 rows

Technical details
Node id
n-056fa906003b296d · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-0fee5d4c7376c95e

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:                 3600,                  5616,                   1296;
  1:               216000,                575424,                 326592,                  46656;
  2:             12960000,              49579776,               48708864,               16236288,                1679616;
  3:            777600000,            3895962624,             5654707200,             3191270400,              745749504,               60466176;
  4:          46656000000,          289458302976,           566527758336,           469298147328,           182124122112,            32651735040,             2176782336;
  5:        2799360000000,        20719850471424,         51549108092928,         57663447810048,         32497909088256,          9534306631680,          1384433565696,            78364164096;
  6:      167961600000000,      1444579835314176,       4385285711659008,       6268448166838272,       4746186548379648,       2002465606533120,        470498441232384,         57362568118272,         2821109907456;
  7:    10077696000000000,     98764062607540224,     355089133600505856,     623653781483814912,     602457424436920320,     339379521866956800,     113735867028996096,      22241630510383104,      2335879003373568,      101559956668416;
  8:   604661760000000000,   6651342763380965376,   27692240540380692480,   58038004521448243200,   68931420764526084096,   49363149232045817856,   21959023004293791744,    6083441404438118400,   1017630765817528320,    93841399961616384,    3656158440062976;
  9: 36279705600000000000, 442613930001143169024, 2096934267535421865984, 5127450099634082414592, 7281487249404170600448, 6399590778883119513600, 3604463609639098908672, 1318639852748953092096, 310729593504072204288, 45380238558061658112, 3729281608864235520, 131621703842267136;
%F Formula
The row polynomials satisfy P(n,x) = -(1440-432*x)*P(n-2,x) + (84+36*x)*P(n-1,x) for n >= 2, with P(0,x) = 3600 + 5616*x + 1296*x^2, P(1,x) = 216000 + 575424*x + 326592*x^2 + 46656*x^3.
%t Mathematica
P[0, x_] := 3600 + 5616*x + 1296*x^2;
P[1, x_] := 216000 + 575424*x + 326592*x^2 + 46656*x^3;
P[n_, x_] := P[n, x] = Expand[-(1440 - 432*x)*P[n - 2, x] + (84 + 36*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3600 + 5616*x + 1296*x^2; v[2] = 216000 + 575424*x + 326592*x^2 + 46656*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(1440 - 432*x)*v[i - 2] + (84 + 36*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

Same fingerprint ?

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.