Real-rooted atlas

A108838 → (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
\(\left(3 - 3\,x + 4\,n - 4\,n x + n^{2} + 5\,n^{2} x + 3\,n^{2} x^{2} + 8\,n^{3} x + 10\,n^{3} x^{2} + 2\,n^{4} x + 12\,n^{4} x^{2} + 6\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(6 - 6\,x + 5\,n + 15\,n x - 11\,n x^{2} + 3\,n x^{3} + n^{2} + 23\,n^{2} x + 16\,n^{2} x^{2} + 16\,n^{2} x^{3} + 10\,n^{3} x + 41\,n^{3} x^{2} + 43\,n^{3} x^{3} + 2\,n^{4} x + 26\,n^{4} x^{2} + 44\,n^{4} x^{3} + 7\,n^{5} x^{2} + 19\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + 3\,x^{3} - 3\,x^{4} + 2\,n x + 8\,n x^{2} + 6\,n x^{3} - 16\,n x^{4} + 12\,n^{2} x^{2} + 23\,n^{2} x^{3} - 35\,n^{2} x^{4} + 4\,n^{3} x^{2} + 28\,n^{3} x^{3} - 32\,n^{3} x^{4} + 13\,n^{4} x^{3} - 13\,n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
13811
24321854
358013532252
46160540384501140
5728015752240875725049
6844837808960700017289822022
796727938282243675018144308712894952
8109601512075264147000120960411605120162405756
9111320267301774084851005987523395708448080192001721590
1012176044550380160138600023950082037420844800160380120002427262332
1113228870785755040353925081544329711702604032019111952860001730328830488616
121429121081081409408825825024463296388468083382579216054038400400048448424192338127481640
131536401597052498496178928756625476013527013515704832010435124739039000787286578624032955392531210645
1416448022932042398723643640016490073642084042062819328055653998429149120088176088146764801230320439044502207002950

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Technical details
Node id
n-d9e6a4222d7d90c1 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2751883146f8224a

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): 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:  2;
  1:  3,    8;
  2:  4,   32,    18;
  3:  5,   80,   135,     32;
  4:  6,  160,   540,    384,     50;
  5:  7,  280,  1575,   2240,    875,     72;
  6:  8,  448,  3780,   8960,   7000,   1728,     98;
  7:  9,  672,  7938,  28224,  36750,  18144,   3087,   128;
  8: 10,  960, 15120,  75264, 147000, 120960,  41160,  5120,  162;
  9: 11, 1320, 26730, 177408, 485100, 598752, 339570, 84480, 8019, 200;
%F Formula
The row polynomials satisfy (n^2+4*n+3+(2*n^4+8*n^3+5*n^2-4*n-3)*x+(n^6+6*n^5+12*n^4+10*n^3+3*n^2)*x^2)*P(n,x) = (n^2+5*n+6+(2*n^4+10*n^3+23*n^2+15*n-6)*x+(n^6+7*n^5+26*n^4+41*n^3+16*n^2-11*n)*x^2+(3*n^6+19*n^5+44*n^4+43*n^3+16*n^2+3*n)*x^3)*P(n-1,x) + ((2*n+3)*x+(4*n^3+12*n^2+8*n-3)*x^2+(2*n^5+13*n^4+28*n^3+23*n^2+6*n+3)*x^3-(2*n^5+13*n^4+32*n^3+35*n^2+16*n+3)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 6 + (2*n^4 + 10*n^3 + 23*n^2 + 15*n - 6)*x + (n^6 + 7*n^5 + 26*n^4 + 41*n^3 + 16*n^2 - 11*n)*x^2 + (3*n^6 + 19*n^5 + 44*n^4 + 43*n^3 + 16*n^2 + 3*n)*x^3)*P[n - 1, x] + ((2*n + 3)*x + (4*n^3 + 12*n^2 + 8*n - 3)*x^2 + (2*n^5 + 13*n^4 + 28*n^3 + 23*n^2 + 6*n + 3)*x^3 - (2*n^5 + 13*n^4 + 32*n^3 + 35*n^2 + 16*n + 3)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + 3 + (2*n^4 + 8*n^3 + 5*n^2 - 4*n - 3)*x + (n^6 + 6*n^5 + 12*n^4 + 10*n^3 + 3*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 6 + (2*n^4 + 10*n^3 + 23*n^2 + 15*n - 6)*x + (n^6 + 7*n^5 + 26*n^4 + 41*n^3 + 16*n^2 - 11*n)*x^2 + (3*n^6 + 19*n^5 + 44*n^4 + 43*n^3 + 16*n^2 + 3*n)*x^3)*v[i - 1] + ((2*n + 3)*x + (4*n^3 + 12*n^2 + 8*n - 3)*x^2 + (2*n^5 + 13*n^4 + 28*n^3 + 23*n^2 + 6*n + 3)*x^3 - (2*n^5 + 13*n^4 + 32*n^3 + 35*n^2 + 16*n + 3)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + 3 + (2*n^4 + 8*n^3 + 5*n^2 - 4*n - 3)*x + (n^6 + 6*n^5 + 12*n^4 + 10*n^3 + 3*n^2)*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-ade1c1d52bd3c716
  • 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-ade1c1d52bd3c716
  • 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-ade1c1d52bd3c716
  • 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-ade1c1d52bd3c716

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023