Real-rooted atlas

A178820 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n + n x\right)\,P_{n}(x) = \left(6 + 21\,x + 15\,x^{2} + 2\,n + 7\,n x + 5\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
182028
240200250490
31601200300025006860
4560560021000350002187584035
5179222400112000280000350000175000941192
653768064050400016800003150000315000013125009882516
71536026880020160008400000210000003150000026250000937500098825160
84224084480073920003696000011550000023100000028875000020625000064453125951192165
911264025344002534400014784000055440000013860000002310000000247500000015468750004296875008877793540
102928647321600823680005491200002402400000720720000015015000000214500000002010937500011171875000279296875080787921214
11745472205004802562560001921920000960960000033633600000840840000001501500000001876875000001564062500007820312500017773437500719746934452
121863680559104007687680006406400000360360000001441440000004204200000009009000000001407656250000156406250000011730468750005332031250001110839843756297785676455
13458752014909440022364160002050048000012812800000057657600000019219200000004804800000000900900000000012512500000000125125000000008531250000000355468750000068359375000054257845827920
1411141120389939200633651200063365120000435635200000217817600000081681600000002333760000000051051000000000850850000000001063562500000009668750000000060429687500000232421875000004150390625000461191689537320
Technical details
Node id
n-aa3c7623db76c626 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9629d1e9c04dbfbe

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:      8,      20;
  2:     40,     200,      250;
  3:    160,    1200,     3000,      2500;
  4:    560,    5600,    21000,     35000,     21875;
  5:   1792,   22400,   112000,    280000,    350000,     175000;
  6:   5376,   80640,   504000,   1680000,   3150000,    3150000,    1312500;
  7:  15360,  268800,  2016000,   8400000,  21000000,   31500000,   26250000,    9375000;
  8:  42240,  844800,  7392000,  36960000, 115500000,  231000000,  288750000,  206250000,   64453125;
  9: 112640, 2534400, 25344000, 147840000, 554400000, 1386000000, 2310000000, 2475000000, 1546875000, 429687500;
%F Formula
The row polynomials satisfy (n+n*x)*P(n,x) = (2*n+6+(7*n+21)*x+(5*n+15)*x^2)*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[((2*n + 6 + (7*n + 21)*x + (5*n + 15)*x^2)*P[n - 1, x])/(n + n*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] = ((2*n + 6 + (7*n + 21)*x + (5*n + 15)*x^2)*v[i - 1])/(n + n*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.