Real-rooted atlas

A196347 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15611
25012072242
37502700324012967986
4150007200012960010368031104351384
5375000225000054000006480000388800093312019326120
61125000081000000243000000388800000349920000167961600335923201275523920
73937500003307500000119070000002381400000028576800000205752960008230118400141087744098215341840

All 15 rows

Technical details
Node id
n-2cf8ae9156c81000 · 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:            5,             6;
  2:           50,           120,             72;
  3:          750,          2700,           3240,            1296;
  4:        15000,         72000,         129600,          103680,           31104;
  5:       375000,       2250000,        5400000,         6480000,         3888000,          933120;
  6:     11250000,      81000000,      243000000,       388800000,       349920000,       167961600,        33592320;
  7:    393750000,    3307500000,    11907000000,     23814000000,     28576800000,     20575296000,      8230118400,     1410877440;
  8:  15750000000,  151200000000,   635040000000,   1524096000000,   2286144000000,   2194698240000,   1316818944000,   451480780800,    67722117120;
  9: 708750000000, 7654500000000, 36741600000000, 102876480000000, 185177664000000, 222213196800000, 177770557440000, 91424858112000, 27427457433600, 3656994324480;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+6*n*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 + 6*n*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 + 6*n*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.