Real-rooted atlas

A280580 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-n + 2\,n^{2}\right)\,P_{n}(x) = \left(30\,x - n - 60\,n x + 2\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 12\,x^{2} + 4\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
31629
411530551
512814014014323
61454201050630422188
716699046206930277213215511
8191200215015420424204212012429113634
91120364040040180180336336240240514801430853467
10115361209282061261218378362450448131274021879048626536382
111190969019380017635807759752166280401662804069283509237801679650852019
121231146303730654476780271591328535727213718133010669659035565530387987658786400763223
1312762125267298010296594823727523569485928413788241051723530654405752178474296162249362080123192727797
14132529900115115021871850223092870127481640041431533007595781050759578105038669430808788507006760390074290025669818476
151378409501883700435134705511706204015671660172100214004350310965063804560820522037315802214703764042590457002808162002674440208023278209
Technical details
Node id
n-e103f8fc18e5e16f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d5ad2f0a26762abe

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   6,    2;
   4: 1,  15,   30,     5;
   5: 1,  28,  140,   140,     14;
   6: 1,  45,  420,  1050,    630,      42;
   7: 1,  66,  990,  4620,   6930,    2772,     132;
   8: 1,  91, 2002, 15015,  42042,   42042,   12012,     429;
   9: 1, 120, 3640, 40040, 180180,  336336,  240240,   51480,   1430;
  10: 1, 153, 6120, 92820, 612612, 1837836, 2450448, 1312740, 218790, 4862;
%F Formula
The row polynomials satisfy (2*n^2-n)*P(n,x) = (2*n^2-n+(24*n^2-60*n+30)*x)*P(n-1,x) + ((4*n-3)*x-(16*n-12)*x^2)*d/dx 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[((2*n^2 - n + (24*n^2 - 60*n + 30)*x)*P[n - 1, x] + ((4*n - 3)*x - (16*n - 12)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 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] = ((2*n^2 - n + (24*n^2 - 60*n + 30)*x)*v[i - 1] + ((4*n - 3)*x - (16*n - 12)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.