Real-rooted atlas

A272866 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 30\,x + 64\,x^{2} + 58\,x^{3} + 19\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15201944
2252005907603611936
312515007425194002821521660685985184
46251000069500274000670150104120010035805487201303213748096
531256250055937529500001015125023815000385747504259800030694025130321002476099164916224
615625375000410625027125000120384375378150000862047500143697000017383503751488403000856208970297131880470458817256313856
77812521875002832812522487500012224406254813812500141605406253161065000053810054375695114525006707776197546889495800224458374356586423340893871739319277809664

All 15 rows

Technical details
Node id
n-8253cbfd755ffdf6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a634a8e2c7c6279b

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       5,       20,         19;
  2:      25,      200,        590,         760,         361;
  3:     125,     1500,       7425,       19400,       28215,        21660,          6859;
  4:     625,    10000,      69500,      274000,      670150,      1041200,       1003580,        548720,         130321;
  5:    3125,    62500,     559375,     2950000,    10151250,     23815000,      38574750,      42598000,       30694025,       13032100,        2476099;
  6:   15625,   375000,    4106250,    27125000,   120384375,    378150000,     862047500,    1436970000,     1738350375,     1488403000,      856208970,       297131880,        47045881;
  7:   78125,  2187500,   28328125,   224875000,  1222440625,   4813812500,   14160540625,   31610650000,    53810054375,    69511452500,    67077761975,     46889495800,     22445837435,      6586423340,      893871739;
  8:  390625, 12500000,  186875000,  1732500000, 11147937500,  52790500000,  190305325000,  532726500000,  1170313543750,  2024360700000,  2748008893000,   2896720316000,   2324496580700,   1372749285600,   562668736760,   143019478240,    16983563041;
  9: 1953125, 70312500, 1191796875, 12637500000, 93940312500, 519828750000, 2219147437500, 7472758500000, 20121898893750, 43649877875000, 76463215796250, 107906632740000, 121769058190500, 108391364046000, 74433764429100, 38050708552800, 13636013378445, 3057041347380, 322687697779;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+30*x+64*x^2+58*x^3+19*x^4)*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 + 30*x + 64*x^2 + 58*x^3 + 19*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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 + 30*x + 64*x^2 + 58*x^3 + 19*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.