Real-rooted atlas

A286785: Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

A286785 ↗ nonnegative coefficients

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

Recurrence from A286785

\(\left(56 + 30\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(156 + 84\,x + 96\,n + 66\,n x + 15\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(2 + 26\,x - 28\,x^{2} + n + 7\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 14 + 14\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01414230
14272272143
2132330220442728
3429143014305206523876
41430600681904550105090221318
548622475243316333201190019041192120175
6167961007762170562170561085282713231921522690690
7587864069801046520130233685465830164455860504018924032015

All 15 rows

Technical details
Node id
n-91d76a88748fc9ba · revision 0 · JSON
Definition source
oeis_catalog:A286785
Fingerprint
f6-ca8fcf8c70f1b4a8

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 ?

Row sums ? A006013, A305574

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     14,      14,        2;
  1:     42,      72,       27,        2;
  2:    132,     330,      220,       44,        2;
  3:    429,    1430,     1430,      520,       65,        2;
  4:   1430,    6006,     8190,     4550,     1050,       90,       2;
  5:   4862,   24752,    43316,    33320,    11900,     1904,     119,       2;
  6:  16796,  100776,   217056,   217056,   108528,    27132,    3192,     152,      2;
  7:  58786,  406980,  1046520,  1302336,   854658,   301644,   55860,    5040,    189,     2;
  8: 208012, 1634380,  4903140,  7354710,  6056820,  2826516,  743820,  106260,   7590,   230,   2;
  9: 742900, 6537520, 22472725, 39657750, 39657750, 23377200, 8182020, 1669800, 189750, 11000, 275, 2;
%F Formula
The row polynomials satisfy (4*n^2+30*n+56)*P(n,x) = (15*n^2+96*n+156+(12*n^2+66*n+84)*x)*P(n-1,x) + (n+2+(7*n+26)*x-(8*n+28)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 14*x + 2*x^2.
%t Mathematica
P[0, x_] := 14 + 14*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n^2 + 96*n + 156 + (12*n^2 + 66*n + 84)*x)*P[n - 1, x] + (n + 2 + (7*n + 26)*x - (8*n + 28)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 30*n + 56)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 14 + 14*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^2 + 96*n + 156 + (12*n^2 + 66*n + 84)*x)*v[i - 1] + (n + 2 + (7*n + 26)*x - (8*n + 28)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 30*n + 56)); 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

Used in 5

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.