Real-rooted atlas

A178821 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(20 + 36\,x + 5\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1254570
2375135012152940
3437523625425252551596040
44375031500085050010206004592702689120
53937503543750127575002296350020667150744017467765824
63281250354375001594687503827250005166787503720087001116026101581202560
725781250324843750175415625052624687509472443750102302392506138143550157837977034786456320

All 15 rows

Technical details
Node id
n-411e78f82a532dd1 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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:         25,          45;
  2:        375,        1350,         1215;
  3:       4375,       23625,        42525,        25515;
  4:      43750,      315000,       850500,      1020600,        459270;
  5:     393750,     3543750,     12757500,     22963500,      20667150,       7440174;
  6:    3281250,    35437500,    159468750,    382725000,     516678750,     372008700,     111602610;
  7:   25781250,   324843750,   1754156250,   5262468750,    9472443750,   10230239250,    6138143550,    1578379770;
  8:  193359375,  2784375000,  17541562500,  63149625000,  142086656250,  204604785000,  184144306500,   94702786200,   21308126895;
  9: 1396484375, 22623046875, 162885937500, 684120937500, 1847126531250, 3324827756250, 3989793307500, 3077840551500, 1385028248175, 277005649635;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+20+(9*n+36)*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[Together[((5*n + 20 + (9*n + 36)*x)*P[n - 1, x])/(n)]];
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 + 20 + (9*n + 36)*x)*v[i - 1])/(n)); 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.