Real-rooted atlas

A034870 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 - 4\,x + x^{2} + 6\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 6\,x^{2} + 4\,x^{3} + x^{4} + 6\,n x + 16\,n x^{2} + 14\,n x^{3} + 4\,n x^{4} + 4\,n^{2} x^{2} + 8\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118918
2116546425160
3124135320375216491120
4132252896175020161372512816912
51404051920525090721029076803645100012139424
6148594352012375285124527650688400952200079861728169212992
715681958242502572072147147219648243243200200121121524161537927442251105920

Show all 15 stored rows

Generated b-files: A002698 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-4333cc412b74398d · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-561a7d876fc231a6

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  8,    9;
  2: 1, 16,   54,    64,    25;
  3: 1, 24,  135,   320,   375,    216,     49;
  4: 1, 32,  252,   896,  1750,   2016,   1372,     512,      81;
  5: 1, 40,  405,  1920,  5250,   9072,  10290,    7680,    3645,    1000,     121;
  6: 1, 48,  594,  3520, 12375,  28512,  45276,   50688,   40095,   22000,    7986,    1728,     169;
  7: 1, 56,  819,  5824, 25025,  72072, 147147,  219648,  243243,  200200,  121121,   52416,   15379,    2744,    225;
  8: 1, 64, 1080,  8960, 45500, 157248, 392392,  732160, 1042470, 1144000,  968968,  628992,  307580,  109760,  27000,   4096,   289;
  9: 1, 72, 1377, 13056, 76500, 308448, 909636, 2036736, 3544398, 4862000, 5294718, 4582656, 3137316, 1679328, 688500, 208896, 44217, 5832, 361;
%F Formula
The row polynomials satisfy (1+(6*n-4)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (1+(6*n+4)*x+(4*n^2+16*n+6)*x^2+(8*n^2+14*n+4)*x^3+(4*n^2+4*n+1)*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[((1 + (6*n + 4)*x + (4*n^2 + 16*n + 6)*x^2 + (8*n^2 + 14*n + 4)*x^3 + (4*n^2 + 4*n + 1)*x^4)*P[n - 1, x])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*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] = ((1 + (6*n + 4)*x + (4*n^2 + 16*n + 6)*x^2 + (8*n^2 + 14*n + 4)*x^3 + (4*n^2 + 4*n + 1)*x^4)*v[i - 1])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-cfa88130c4a396cc
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-cfa88130c4a396cc
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-cfa88130c4a396cc

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.