Real-rooted atlas

A391533 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(4 + 12\,x + 21\,x^{2} + 15\,x^{3} + 4\,x^{4} + 12\,n x + 20\,n x^{2} + 15\,n x^{3} + 4\,n x^{4} + 4\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n}(x) = \left(4 + 28\,x + 69\,x^{2} + 79\,x^{3} + 43\,x^{4} + 9\,x^{5} + 12\,n x + 40\,n x^{2} + 51\,n x^{3} + 29\,n x^{4} + 6\,n x^{5} + 4\,n^{2} x^{2} + 8\,n^{2} x^{3} + 5\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 4 + 16\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0416929
1432451697
24481179625290
346422530417536808
4480369704650288492144
5496549136017501224441645488
6411276523363875374421076408113664
7412810173696752593247203339289110033280
8414413055504133002016019894128005184120012179616
9416016297824219003931247334387842138476001573144187648
10417619891072034125708481008421006087047034000107692016169436736
11419223851425650875119988197274232320197802121000519091483225351961005568
12420828171849673150193248360591489984491832365200198319766081994231362252293760
13422432852350410205029858462362396096011119689724006402113126241098502626438252565189632
14424037892934413877544553610300291775488232818323452001816815107452847674915366433975460828911657216

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Technical details
Node id
n-cd2d00ce01d1d8bf · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b6fea4e2006d7377

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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 ?

No OEIS matches.

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: 4,  16,    9;
  1: 4,  32,   45,   16;
  2: 4,  48,  117,   96,    25;
  3: 4,  64,  225,  304,   175,    36;
  4: 4,  80,  369,  704,   650,   288,    49;
  5: 4,  96,  549, 1360,  1750,  1224,   441,    64;
  6: 4, 112,  765, 2336,  3875,  3744,  2107,   640,    81;
  7: 4, 128, 1017, 3696,  7525,  9324,  7203,  3392,   891,  100;
  8: 4, 144, 1305, 5504, 13300, 20160, 19894, 12800,  5184, 1200,  121;
  9: 4, 160, 1629, 7824, 21900, 39312, 47334, 38784, 21384, 7600, 1573, 144;
%F Formula
The row polynomials satisfy (4+(12*n+12)*x+(4*n^2+20*n+21)*x^2+(4*n^2+15*n+15)*x^3+(n^2+4*n+4)*x^4)*P(n,x) = (4+(12*n+28)*x+(4*n^2+40*n+69)*x^2+(8*n^2+51*n+79)*x^3+(5*n^2+29*n+43)*x^4+(n^2+6*n+9)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 16*x + 9*x^2.
%t Mathematica
P[0, x_] := 4 + 16*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4 + (12*n + 28)*x + (4*n^2 + 40*n + 69)*x^2 + (8*n^2 + 51*n + 79)*x^3 + (5*n^2 + 29*n + 43)*x^4 + (n^2 + 6*n + 9)*x^5)*P[n - 1, x])/(4 + (12*n + 12)*x + (4*n^2 + 20*n + 21)*x^2 + (4*n^2 + 15*n + 15)*x^3 + (n^2 + 4*n + 4)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4 + 16*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + (12*n + 28)*x + (4*n^2 + 40*n + 69)*x^2 + (8*n^2 + 51*n + 79)*x^3 + (5*n^2 + 29*n + 43)*x^4 + (n^2 + 6*n + 9)*x^5)*v[i - 1])/(4 + (12*n + 12)*x + (4*n^2 + 20*n + 21)*x^2 + (4*n^2 + 15*n + 15)*x^3 + (n^2 + 4*n + 4)*x^4)); 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-247d5155af4042c4
  • 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-247d5155af4042c4
  • 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-247d5155af4042c4
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-247d5155af4042c4

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