Real-rooted atlas

A038334 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(40 + 32\,x + 20\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 640 + 512\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
06405121152
119200307201228862208
251200012288009830402621442985984
3128000004096000049152000262144005242880134369280
43072000001228800000196608000015728640006291456001006632965804752896
5716800000034406400000688128000007340032000044040192000140928614401879048192243799621632
616384000000091750400000022020096000002936012800000234881024000011274289152003006477107203435973836810030613004288
73686400000000235929600000006606028800000010569646080000010569646080000067645734912000270582939648006184752906240618475290624406239826673664

All 15 rows

Technical details
Node id
n-e46ced92cad8408a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6b726760bd05165a

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:              640,               512;
  1:            19200,             30720,             12288;
  2:           512000,           1228800,            983040,             262144;
  3:         12800000,          40960000,          49152000,           26214400,            5242880;
  4:        307200000,        1228800000,        1966080000,         1572864000,          629145600,          100663296;
  5:       7168000000,       34406400000,       68812800000,        73400320000,        44040192000,        14092861440,        1879048192;
  6:     163840000000,      917504000000,     2202009600000,      2936012800000,      2348810240000,      1127428915200,      300647710720,       34359738368;
  7:    3686400000000,    23592960000000,    66060288000000,    105696460800000,    105696460800000,     67645734912000,    27058293964800,     6184752906240,      618475290624;
  8:   81920000000000,   589824000000000,  1887436800000000,   3523215360000000,   4227858432000000,   3382286745600000,  1803886264320000,   618475290624000,   123695058124800,   10995116277760;
  9: 1802240000000000, 14417920000000000, 51904512000000000, 110729625600000000, 155021475840000000, 148820616806400000, 99213744537600000, 45354854645760000, 13606456393728000, 2418925581107200, 193514046488576;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (20*n+40+(16*n+32)*x)*P(n-1,x) for n >= 1, with P(0,x) = 640 + 512*x.
%t Mathematica
P[0, x_] := 640 + 512*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (16*n + 32)*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] = 640 + 512*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n + 40 + (16*n + 32)*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.