Real-rooted atlas

A145597 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(6\,n + 12\,n x + 6\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(12 + 48\,x + 60\,x^{2} + 24\,x^{3} + 8\,n + 32\,n x + 40\,n x^{2} + 16\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
26272760
31075165110360
4151655948584292061
521315163838224095163811529
6285463822127402184018564618863728
736882793835280856801156688139623256350136
84513501512085680275400523260581400348840872101918305
955198026928188496767448191862029845202813976147094232687610499841
1066280545441383724191862060436531231114516180362132384786128925122578557479004
117838617335973359044015401694592943306263742393088436285060928725253088554601610314905968
129151871141141331330941440543306263134730596288708420426501075426501075275585310103791870172986451727288381
1310568251719902312310189939751025674653798795009876867001818241425235697962521031510501229114250423361575651325509487599345
141208820252252386786436468432227927700987686700304771896067881013201093638546012618906300101606778005419028160171949932024564276052192171968

Computed in your browser ?

Technical details
Node id
n-1a7a27d6b2026223 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-663a3c2c5c184f3a

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,    6;
  2:  6,   27,    27;
  3: 10,   75,   165,    110;
  4: 15,  165,   594,    858,    429;
  5: 21,  315,  1638,   3822,   4095,    1638;
  6: 28,  546,  3822,  12740,  21840,   18564,    6188;
  7: 36,  882,  7938,  35280,  85680,  115668,   81396,   23256;
  8: 45, 1350, 15120,  85680, 275400,  523260,  581400,  348840,   87210;
  9: 55, 1980, 26928, 188496, 767448, 1918620, 2984520, 2813976, 1470942, 326876;
%F Formula
The row polynomials satisfy (n^2+6*n+(2*n^2+12*n)*x+(n^2+6*n)*x^2)*P(n,x) = (n^2+8*n+12+(4*n^2+32*n+48)*x+(5*n^2+40*n+60)*x^2+(2*n^2+16*n+24)*x^3)*P(n-1,x) + ((2*n+4)*x+(6*n+12)*x^2+(6*n+12)*x^3+(2*n+4)*x^4)*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[((n^2 + 8*n + 12 + (4*n^2 + 32*n + 48)*x + (5*n^2 + 40*n + 60)*x^2 + (2*n^2 + 16*n + 24)*x^3)*P[n - 1, x] + ((2*n + 4)*x + (6*n + 12)*x^2 + (6*n + 12)*x^3 + (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^2 + 6*n + (2*n^2 + 12*n)*x + (n^2 + 6*n)*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] = ((n^2 + 8*n + 12 + (4*n^2 + 32*n + 48)*x + (5*n^2 + 40*n + 60)*x^2 + (2*n^2 + 16*n + 24)*x^3)*v[i - 1] + ((2*n + 4)*x + (6*n + 12)*x^2 + (6*n + 12)*x^3 + (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^2 + 6*n + (2*n^2 + 12*n)*x + (n^2 + 6*n)*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 [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-8d309dafbf46f610
  • 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-8d309dafbf46f610
  • 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-8d309dafbf46f610

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023