Real-rooted atlas

A156992 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 6\,x + 5\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1101222
2150360216726
330001080012960518431944
4750003600006480005184001555201756920
52250000135000003240000038880000233280005598720115956720
67875000056700000017010000002721600000244944000011757312002351462408928667440
7315000000026460000000952560000001905120000002286144000001646023680006584094720011287019520785722734720

All 15 rows

Technical details
Node id
n-3992e0c998ec014d · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             1;
  1:            10,             12;
  2:           150,            360,             216;
  3:          3000,          10800,           12960,             5184;
  4:         75000,         360000,          648000,           518400,           155520;
  5:       2250000,       13500000,        32400000,         38880000,         23328000,          5598720;
  6:      78750000,      567000000,      1701000000,       2721600000,       2449440000,       1175731200,        235146240;
  7:    3150000000,    26460000000,     95256000000,     190512000000,     228614400000,     164602368000,      65840947200,     11287019520;
  8:  141750000000,  1360800000000,   5715360000000,   13716864000000,   20575296000000,   19752284160000,   11851370496000,   4063327027200,    609499054080;
  9: 7087500000000, 76545000000000, 367416000000000, 1028764800000000, 1851776640000000, 2222131968000000, 1777705574400000, 914248581120000, 274274574336000, 36569943244800;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+5+(6*n+6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5*n + 5 + (6*n + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5*n + 5 + (6*n + 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

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.