Real-rooted atlas

A256890 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-12 - 6\,n - 8\,n x - 2\,n x^{2} - 4\,n^{2} x - 3\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 2\,x^{2} + 2\,x^{3} - 2\,n x - n x^{2} + 3\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(7 + 2\,x + 5\,n + 4\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x - x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 12 + 8\,x\)
\(P_{2}(x) = 52 + 104\,x + 24\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
112820
25210424180
3196832588641680
468452887932273616016800
5227629360793205872011380384181440
67340149304660840881120373260440408962116800
72317271465648718921057344081198202143968162204204826611200
87207632757203293533210891163213613954065870664114650205766084608359251200

All 15 rows

Technical details
Node id
n-09a65b728d1d9bb3 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-96547810142a253d

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 0)
  • 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:     12,        8;
   2:     52,      104,         24;
   3:    196,      832,        588,         64;
   4:    684,     5288,       7932,       2736,         160;
   5:   2276,    29360,      79320,      58720,       11380,         384;
   6:   7340,   149304,     660840,     881120,      373260,       44040,         896;
   7:  23172,   714656,    4871892,   10573440,     8119820,     2143968,      162204,       2048;
   8:  72076,  3275720,   32935332,  108911632,   136139540,    65870664,    11465020,     576608,      4608;
   9: 221860, 14544400,  208898880, 1004780000,  1905953560,  1507170000,   487430720,   58177600,   1996740,   10240;
  10: 677868, 63058520, 1262660400, 8535466560, 23389474920, 28067369904, 14937066480, 3367094400, 283763340, 6778680, 22528;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(6*n+12+(4*n^2+8*n)*x+(n^3+3*n^2+2*n)*x^2)*P(n-2,x) - ((2*n+4)*x+(n^2+n-2)*x^2-(n^2+3*n+2)*x^3)*d/dx P(n-2,x) + (5*n+7+(2*n^2+4*n+2)*x)*P(n-1,x) + ((n+1)*x-(n+1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 8*x, P(2,x) = 52 + 104*x + 24*x^2.
%t Mathematica
P[1, x_] := 12 + 8*x;
P[2, x_] := 52 + 104*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n + 12 + (4*n^2 + 8*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*P[n - 2, x] - ((2*n + 4)*x + (n^2 + n - 2)*x^2 - (n^2 + 3*n + 2)*x^3)*D[P[n - 2, x], x] + (5*n + 7 + (2*n^2 + 4*n + 2)*x)*P[n - 1, x] + ((n + 1)*x - (n + 1)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 12 + 8*x; v[2] = 52 + 104*x + 24*x^2; for(n = 3, N, my(i = n); v[i] = (-(6*n + 12 + (4*n^2 + 8*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*v[i - 2] - ((2*n + 4)*x + (n^2 + n - 2)*x^2 - (n^2 + 3*n + 2)*x^3)*deriv(v[i - 2], x) + (5*n + 7 + (2*n^2 + 4*n + 2)*x)*v[i - 1] + ((n + 1)*x - (n + 1)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.