A165253 → D+1

OEIS matches ?

• A038731: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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.

Triangle begins:
  0:  1;
  1:  1;
  2:  2,   1;
  3:  4,   5,    1;
  4:  7,  16,    8,    1;
  5: 11,  40,   36,   11,    1;
  6: 16,  85,  119,   64,   14,   1;
  7: 22, 161,  322,  264,  100,  17,   1;
  8: 29, 280,  756,  870,  495, 144,  20,  1;
  9: 37, 456, 1596, 2442, 1925, 832, 196, 23, 1;

The row polynomials satisfy (2*n^3-9*n^2+17*n-12-(4*n-6)*x-(2*n-3)*x^2)*P(n,x) = (2*n^3-5*n^2+7*n-6+(n^3-2*n^2-5*n+8)*x-(5*n-6)*x^2-(n-1)*x^3)*P(n-1,x) + ((4*n^2-8*n-4)*x+(n^2-2*n-17)*x^2-8*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 5*n^2 + 7*n - 6 + (n^3 - 2*n^2 - 5*n + 8)*x - (5*n - 6)*x^2 - (n - 1)*x^3)*P[n - 1, x] + ((4*n^2 - 8*n - 4)*x + (n^2 - 2*n - 17)*x^2 - 8*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 9*n^2 + 17*n - 12 - (4*n - 6)*x - (2*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 - 5*n^2 + 7*n - 6 + (n^3 - 2*n^2 - 5*n + 8)*x - (5*n - 6)*x^2 - (n - 1)*x^3)*v[i - 1] + ((4*n^2 - 8*n - 4)*x + (n^2 - 2*n - 17)*x^2 - 8*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 9*n^2 + 17*n - 12 - (4*n - 6)*x - (2*n - 3)*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)

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-05447d659c882bb2
• 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-05447d659c882bb2
• 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-05447d659c882bb2
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-05447d659c882bb2, e-2e28139872aa1c07
• Real-rooted. Every row P_n, n >= 1, 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-2e28139872aa1c07
• Coefficients of one sign. Every nonzero row, n >= 1: 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-2e28139872aa1c07
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-2e28139872aa1c07

Proof chains

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

• preserver_image derivative_polynomial, shift 0 e-05447d659c882bb2

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-2580795600db03bb root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • theorem_check three_term_recurrence e-365c9804d88a701d root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

• preserver_image derivative_polynomial, shift 0 e-2e28139872aa1c07

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-cec419fd822ca749 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • theorem_check three_term_recurrence e-59d957f5dca11392 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

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