A078812 → D+1

OEIS matches ?

• A030267: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:
   1:   1;
   2:   3,    1;
   3:   7,    6,    1;
   4:  14,   22,    9,    1;
   5:  25,   62,   45,   12,    1;
   6:  41,  147,  164,   76,   15,    1;
   7:  63,  308,  486,  340,  115,   18,    1;
   8:  92,  588, 1242, 1210,  610,  162,   21,   1;
   9: 129, 1044, 2838, 3652, 2535,  994,  217,  24,  1;
  10: 175, 1749, 5940, 9724, 8827, 4725, 1512, 280, 27, 1;

The row polynomials satisfy (2*n^3-6*n^2+16*n-12-(2*n-2)*x-(2*n-2)*x^2)*P(n,x) = (2*n^3+10*n+(n^3-n)*x-4*n*x^2-n*x^3)*P(n-1,x) + ((4*n^2-4*n+8)*x+(n^2-n-10)*x^2-7*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 10*n + (n^3 - n)*x - 4*n*x^2 - n*x^3)*P[n - 1, x] + ((4*n^2 - 4*n + 8)*x + (n^2 - n - 10)*x^2 - 7*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 16*n - 12 - (2*n - 2)*x - (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^3 + 10*n + (n^3 - n)*x - 4*n*x^2 - n*x^3)*v[i - 1] + ((4*n^2 - 4*n + 8)*x + (n^2 - n - 10)*x^2 - 7*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + 16*n - 12 - (2*n - 2)*x - (2*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, v[i])))};
T(10)

Proved properties

• 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-7a38528e1fbfc148
• 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-7a38528e1fbfc148
• 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-7a38528e1fbfc148
• 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-7a38528e1fbfc148

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

• preserver_image derivative_polynomial, shift 0 e-7a38528e1fbfc148

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

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-537ef91063de36e5 root: theorem check

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

    • theorem_check three_term_recurrence e-5aa1c8cf88fab58c 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