Real-rooted atlas

A159854 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 16 + 48\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0164836100
1803364682161100
2400216043563888129612100
32000132003474045576298087776133100
41000078000252900436320422496217728466561464100
5500004500001732500369900047304003623616153964827993616105100
625000025500001136250028890000458460004650048029439936106375681679616177156100
71250000142500007211250021262500040257000050757840042620256022982745672223488100776961948717100

All 15 rows

Technical details
Node id
n-215e8306d8f3616b · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-7b197100572898ab

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       16,        48,         36;
  1:       80,       336,        468,         216;
  2:      400,      2160,       4356,        3888,        1296;
  3:     2000,     13200,      34740,       45576,       29808,        7776;
  4:    10000,     78000,     252900,      436320,      422496,      217728,       46656;
  5:    50000,    450000,    1732500,     3699000,     4730400,     3623616,     1539648,      279936;
  6:   250000,   2550000,   11362500,    28890000,    45846000,    46500480,    29439936,    10637568,     1679616;
  7:  1250000,  14250000,   72112500,   212625000,   402570000,   507578400,   426202560,   229827456,    72223488,    10077696;
  8:  6250000,  78750000,  446062500,  1495800000,  3288600000,  4953312000,  5176483200,  3706352640,  1740082176,   483729408,   60466176;
  9: 31250000, 431250000, 2702812500, 10155375000, 25417800000, 44498160000, 55602288000, 49590662400, 30938526720, 12859140096, 3204707328, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (5+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + 48*x + 36*x^2.
%t Mathematica
P[0, x_] := 16 + 48*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 16 + 48*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 6*x)*v[i - 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.