Real-rooted atlas

A038301 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(20 + 62\,x + 42\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1680\,x + 1764\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0400168017643844
180005040010584074088238328
2160000134400042336005927040311169614776336
3320000033600000141120000296352000311169600130691232916132832
4640000008064000004233600000118540800001867017600015682947840548903174456800235584
512800000001881600000011854080000041489280000087127488000010978063488007684644441602305393332483521614606208
625600000000430080000000316108800000013276569600000348509952000005854967193600061477155532800368862933196809682651996416218340105584896
75120000000009676800000000812851200000003982970880000001254635827200000263473523712000036886293319680003319766398771200174287735935488040667138384947213537086546263552

All 15 rows

Technical details
Node id
n-45aa4f29cc8bfb73 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-09bb967aa8554943

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:             400,             1680,              1764;
  1:            8000,            50400,            105840,              74088;
  2:          160000,          1344000,           4233600,            5927040,             3111696;
  3:         3200000,         33600000,         141120000,          296352000,           311169600,           130691232;
  4:        64000000,        806400000,        4233600000,        11854080000,         18670176000,         15682947840,          5489031744;
  5:      1280000000,      18816000000,      118540800000,       414892800000,        871274880000,       1097806348800,        768464444160,         230539333248;
  6:     25600000000,     430080000000,     3161088000000,     13276569600000,      34850995200000,      58549671936000,      61477155532800,       36886293319680,        9682651996416;
  7:    512000000000,    9676800000000,    81285120000000,    398297088000000,    1254635827200000,    2634735237120000,    3688629331968000,     3319766398771200,     1742877359354880,     406671383849472;
  8:  10240000000000,  215040000000000,  2032128000000000,  11379916800000000,   41821194240000000,  105389409484800000,  184431466598400000,   221317759918080000,   174287735935488000,   81334276769894400,   17080198121677824;
  9: 204800000000000, 4730880000000000, 49674240000000000, 312947712000000000, 1314380390400000000, 3864278347776000000, 8114984530329600000, 12172476795494400000, 12781100635269120000, 8946770444688384000, 3757643586769121280, 717368321110468608;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (20+62*x+42*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1680*x + 1764*x^2.
%t Mathematica
P[0, x_] := 400 + 1680*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((20 + 62*x + 42*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1680*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((20 + 62*x + 42*x^2)*v[i - 1])/(1 + x)); 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.