Real-rooted atlas

A038284 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(38 + 24\,x\right)\,P_{n-1}(x) + \left(24\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1444 + 2400\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0144424005764420
15487218345610713613824359288
220851361269120013616640409190433177632816656
379235168836897664147561984077690880014266368079626243319287776
4301093638453789299200146988850176120874844160377627443204682022912191102976367299800128
51144155825923407199023616139319842959361682396526182479612039987201646065483776147149291520458647142444036569409408
643477921384962139923134464001279923426017280218500380190310414705765021122564511472424058886628674266726444763961098241100753141765675864292114688
7165216101262848133718704450007041152092301601996802710685804337561602494973425287168001065751003508244482289172098318336025130194226380801327508288962562641807540224781427473062018560

All 15 rows

Technical details
Node id
n-4517c75d35b86f26 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c047d6bccc92b55f

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:               1444,                 2400,                   576;
  1:              54872,               183456,                107136,                  13824;
  2:            2085136,             12691200,              13616640,                4091904,                 331776;
  3:           79235168,            836897664,            1475619840,              776908800,              142663680,                7962624;
  4:         3010936384,          53789299200,          146988850176,           120874844160,            37762744320,             4682022912,              191102976;
  5:       114415582592,        3407199023616,        13931984295936,         16823965261824,          7961203998720,          1646065483776,           147149291520,            4586471424;
  6:      4347792138496,      213992313446400,      1279923426017280,       2185003801903104,       1470576502112256,        451147242405888,         66286742667264,         4476396109824,         110075314176;
  7:    165216101262848,    13371870445000704,    115209230160199680,     271068580433756160,     249497342528716800,     106575100350824448,      22891720983183360,      2513019422638080,      132750828896256,       2641807540224;
  8:   6278211847988224,   833021154020352000,  10228918684457189376,   32582565371557969920,   39938289829258199040,   22826802076119465984,    6724095627359158272,   1067083304659845120,    90845156789452800,    3857039008727040,     63403380965376;
  9: 238572050223552512, 51797988633613541376, 899679514559806734336, 3829576239298349236224, 6133712406037989949440, 4565153683929072402432, 1771628654006233989120, 381197455816547893248, 46504385373410426880, 3159971671163535360, 110195076117823488, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (38+24*x)*P(n-1,x) + 24*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1444 + 2400*x + 576*x^2.
%t Mathematica
P[0, x_] := 1444 + 2400*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(38 + 24*x)*P[n - 1, x] + 24*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] = 1444 + 2400*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (38 + 24*x)*v[i - 1] + 24*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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.