Real-rooted atlas

A085880 → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-30 - 72\,x + 20\,n + 48\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
251217
350240288578
4625450010800864024565
58750840003024004838402903041169294
61312501575000756000018144000217728001045094459633994
720625002970000017820000057024000010264320009853747203941498883186159108
83351562556306250040540500001621620000038918880000560431872004483454976015371845632176035290717

All 15 rows

Technical details
Node id
n-d073c7e23a70d5a1 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f055f948fcf55643

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          1;
   2:          5,           12;
   3:         50,          240,           288;
   4:        625,         4500,         10800,           8640;
   5:       8750,        84000,        302400,         483840,         290304;
   6:     131250,      1575000,       7560000,       18144000,       21772800,       10450944;
   7:    2062500,     29700000,     178200000,      570240000,     1026432000,      985374720,       394149888;
   8:   33515625,    563062500,    4054050000,    16216200000,    38918880000,    56043187200,     44834549760,     15371845632;
   9:  558593750,  10725000000,   90090000000,   432432000000,  1297296000000,  2490808320000,   2988969984000,   2049579417600,   614873825280;
  10: 9496093750, 205115625000, 1969110000000, 11027016000000, 39697257600000, 95273418240000, 152437469184000, 156792825446400, 94075695267840, 25086852071424;
%F Formula
The row polynomials satisfy n*P(n,x) = (20*n-30+(48*n-72)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((20*n - 30 + (48*n - 72)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((20*n - 30 + (48*n - 72)*x)*v[i - 1])/(n)); 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

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.