A054142 → D+1

OEIS matches ?

• A027989: row sums, 15 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:  2,   1;
  2:  4,   5,    1;
  3:  6,  17,    9,    1;
  4:  8,  37,   45,   14,    1;
  5: 10,  65,  133,   95,   20,    1;
  6: 12, 101,  297,  364,  175,   27,    1;
  7: 14, 145,  561, 1005,  840,  294,   35,   1;
  8: 16, 197,  949, 2266, 2805, 1722,  462,  44,  1;
  9: 18, 257, 1485, 4459, 7436, 6831, 3234, 690, 54, 1;

The row polynomials satisfy (8*n^3-20*n^2+16*n-4+(16*n^2-34*n+13)*x+(8*n-4)*x^2)*P(n,x) = (8*n^3-12*n^2+4*n+(24*n^3-36*n^2-8*n+10)*x+(48*n^2-86*n+41)*x^2+(24*n-20)*x^3)*P(n-1,x) - ((4*n^2-2*n)*x+(16*n^2-5)*x^2+(32*n-16)*x^3+16*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[((8*n^3 - 12*n^2 + 4*n + (24*n^3 - 36*n^2 - 8*n + 10)*x + (48*n^2 - 86*n + 41)*x^2 + (24*n - 20)*x^3)*P[n - 1, x] - ((4*n^2 - 2*n)*x + (16*n^2 - 5)*x^2 + (32*n - 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^2 - 34*n + 13)*x + (8*n - 4)*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] = ((8*n^3 - 12*n^2 + 4*n + (24*n^3 - 36*n^2 - 8*n + 10)*x + (48*n^2 - 86*n + 41)*x^2 + (24*n - 20)*x^3)*v[i - 1] - ((4*n^2 - 2*n)*x + (16*n^2 - 5)*x^2 + (32*n - 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^2 - 34*n + 13)*x + (8*n - 4)*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-f4e836bf3f81ac68
• 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-f4e836bf3f81ac68
• 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-f4e836bf3f81ac68
• 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-f4e836bf3f81ac68

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-f4e836bf3f81ac68

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

  • premise (sequence): Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-abbe6e2e6a5c7279 root: theorem check

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

    • gamma_polynomial scale 1, shift 0 e-313e6d9276f9a630

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • basis_map A008288 e-80180a24956ab79f

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-a8f2d1946d8f3a6f

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-054f97e0823e3ab6

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-912230b718f958a6

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement

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