Real-rooted atlas

A121574 → D+1

nonnegative coefficients

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

Recurrence
\(\left(28\,n x - 15\,n x^{2} + 3\,n x^{3} + n x^{4} + 6\,n^{2} - 11\,n^{2} x + 5\,n^{2} x^{2} + n^{2} x^{3} - 6\,n^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(-28\,x^{2} + 20\,x^{3} + 10\,x^{4} + x^{5} + 12\,n + 70\,n x + 41\,n x^{2} - 14\,n x^{3} - 3\,n x^{4} + 2\,n^{2} x - 23\,n^{2} x^{2} - 4\,n^{2} x^{3} - 12\,n^{3} - 12\,n^{3} x\right)\,P_{n-1}(x) + \left(28\,x^{3} + 60\,x^{4} + 15\,x^{5} + x^{6} + 12\,n x + 52\,n x^{2} + 63\,n x^{3} + 15\,n x^{4} + n x^{5} + 12\,n^{2} x + 24\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
297117
3243211168
46011870151264
51443803351231911002
6336112013397291912313739
7768310447363504135427427113768
81728822415352145567631226437231150159
9384021056466405436036547146653513485351181142
1084485248013480018729615509379727257405155613391649392
11184321280003745286060485997523798401568134217972447564312313636
123993630668810078721865088215526816330108300402848696549498349144718199061
13860167239682641664550963273001926468340393385516602954859549738612979108751128921420
14184320168755267724801573248023552864239676801705669586096653094570787770139745167331275551101603885

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Technical details
Node id
n-2ffb0ff2c4cf68d9 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-422d7b1e2dc04ac9

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:    1;
  1:    3,     1;
  2:    9,     7,     1;
  3:   24,    32,    11,     1;
  4:   60,   118,    70,    15,     1;
  5:  144,   380,   335,   123,    19,     1;
  6:  336,  1120,  1339,   729,   191,    23,    1;
  7:  768,  3104,  4736,  3504,  1354,   274,   27,   1;
  8: 1728,  8224, 15352, 14556,  7631,  2264,  372,  31,  1;
  9: 3840, 21056, 46640, 54360, 36547, 14665, 3513, 485, 35, 1;
%F Formula
The row polynomials satisfy -(6*n^3-6*n^2-(3*n^3-11*n^2+28*n)*x-(5*n^2-15*n)*x^2-(n^2+3*n)*x^3-n*x^4)*P(n,x) = -(12*n^3-12*n+(12*n^3-2*n^2-70*n)*x+(23*n^2-41*n+28)*x^2+(4*n^2+14*n-20)*x^3+(3*n-10)*x^4-x^5)*P(n-1,x) + ((12*n^2+12*n)*x+(24*n^2+52*n)*x^2+(3*n^2+63*n+28)*x^3+(15*n+60)*x^4+(n+15)*x^5+x^6)*d/dx 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[(-(12*n^3 - 12*n + (12*n^3 - 2*n^2 - 70*n)*x + (23*n^2 - 41*n + 28)*x^2 + (4*n^2 + 14*n - 20)*x^3 + (3*n - 10)*x^4 - x^5)*P[n - 1, x] + ((12*n^2 + 12*n)*x + (24*n^2 + 52*n)*x^2 + (3*n^2 + 63*n + 28)*x^3 + (15*n + 60)*x^4 + (n + 15)*x^5 + x^6)*D[P[n - 1, x], x])/(-6*n^3 + 6*n^2 + (3*n^3 - 11*n^2 + 28*n)*x + (5*n^2 - 15*n)*x^2 + (n^2 + 3*n)*x^3 + n*x^4)]];
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] = (-(12*n^3 - 12*n + (12*n^3 - 2*n^2 - 70*n)*x + (23*n^2 - 41*n + 28)*x^2 + (4*n^2 + 14*n - 20)*x^3 + (3*n - 10)*x^4 - x^5)*v[i - 1] + ((12*n^2 + 12*n)*x + (24*n^2 + 52*n)*x^2 + (3*n^2 + 63*n + 28)*x^3 + (15*n + 60)*x^4 + (n + 15)*x^5 + x^6)*deriv(v[i - 1], x))/(-6*n^3 + 6*n^2 + (3*n^3 - 11*n^2 + 28*n)*x + (5*n^2 - 15*n)*x^2 + (n^2 + 3*n)*x^3 + n*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-b8ffc6ab03d37d48
  • 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-b8ffc6ab03d37d48
  • 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-b8ffc6ab03d37d48
  • 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-b8ffc6ab03d37d48

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