Real-rooted atlas

A139548 → 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

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(8 + 56\,x + 138\,x^{2} + 140\,x^{3} + 50\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18405098
2646402400400025009604
3512768048000160000300000300000125000941192
4409681920716800358400011200000224000002800000020000000625000092236816
53276881920092160006144000026880000080640000016800000002400000000225000000012500000003125000009039207968
62621447864320108134400901120000506880000020275200000591360000001267200000001980000000002200000000001650000000007500000000015625000000885842380864
72097152734003201192755200119275520008200192000041000960000015375360000004392960000000960960000000016016000000000200200000000001820000000000011375000000000437500000000078125000000086812553324672

All 15 rows

Technical details
Node id
n-48b0227ab432072e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b2b1f354adf1851f

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,         40,           50;
  2:        64,        640,         2400,          4000,           2500;
  3:       512,       7680,        48000,        160000,         300000,          300000,          125000;
  4:      4096,      81920,       716800,       3584000,       11200000,        22400000,        28000000,         20000000,          6250000;
  5:     32768,     819200,      9216000,      61440000,      268800000,       806400000,      1680000000,       2400000000,       2250000000,        1250000000,         312500000;
  6:    262144,    7864320,    108134400,     901120000,     5068800000,     20275200000,     59136000000,     126720000000,     198000000000,      220000000000,      165000000000,        75000000000,        15625000000;
  7:   2097152,   73400320,   1192755200,   11927552000,    82001920000,    410009600000,   1537536000000,    4392960000000,    9609600000000,    16016000000000,    20020000000000,     18200000000000,     11375000000000,      4375000000000,       781250000000;
  8:  16777216,  671088640,  12582912000,  146800640000,  1192755200000,   7156531200000,  32800768000000,  117145600000000,  329472000000000,   732160000000000,  1281280000000000,   1747200000000000,   1820000000000000,   1400000000000000,    750000000000000,    250000000000000,    39062500000000;
  9: 134217728, 6039797760, 128345702400, 1711276032000, 16043212800000, 112302489600000, 608305152000000, 2607022080000000, 8961638400000000, 24893440000000000, 56010240000000000, 101836800000000000, 148512000000000000, 171360000000000000, 153000000000000000, 102000000000000000, 47812500000000000, 14062500000000000, 1953125000000000;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (8+56*x+138*x^2+140*x^3+50*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[((8 + 56*x + 138*x^2 + 140*x^3 + 50*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] = ((8 + 56*x + 138*x^2 + 140*x^3 + 50*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

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