Real-rooted atlas

A038319 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(32 + 20\,x + 16\,n + 10\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 320 + 200\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0320200520
176809600300020280
216384030720019200040000703040
3327680081920007680000320000050000022848800
462914560196608000245760000153600000480000006000000712882560
5117440512044040192006881280000573440000026880000006720000007000000021624104320
62147483648093952409600176160768000183500800000114688000000430080000008960000000800000000642544814080
738654705664019327352832004227858432000528482304000041287680000002064384000000645120000000115200000000900000000018794435811840

All 15 rows

Technical details
Node id
n-729eaa8d4ea8435a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4b28493176e02e0b

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 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             320,             200;
  1:            7680,            9600,             3000;
  2:          163840,          307200,           192000,            40000;
  3:         3276800,         8192000,          7680000,          3200000,           500000;
  4:        62914560,       196608000,        245760000,        153600000,         48000000,          6000000;
  5:      1174405120,      4404019200,       6881280000,       5734400000,       2688000000,        672000000,         70000000;
  6:     21474836480,     93952409600,     176160768000,     183500800000,     114688000000,      43008000000,       8960000000,       800000000;
  7:    386547056640,   1932735283200,    4227858432000,    5284823040000,    4128768000000,    2064384000000,     645120000000,    115200000000,      9000000000;
  8:   6871947673600,  38654705664000,   96636764160000,  140928614400000,  132120576000000,   82575360000000,   34406400000000,   9216000000000,   1440000000000,   100000000000;
  9: 120946279055360, 755914244096000, 2126008811520000, 3543348019200000, 3875536896000000, 2906652672000000, 1513881600000000, 540672000000000, 126720000000000, 17600000000000, 1100000000000;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (16*n+32+(10*n+20)*x)*P(n-1,x) for n >= 1, with P(0,x) = 320 + 200*x.
%t Mathematica
P[0, x_] := 320 + 200*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 32 + (10*n + 20)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 320 + 200*x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 32 + (10*n + 20)*x)*v[i - 1])/(n + 1)); 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

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.