A091044 → D+1

OEIS matches ?

• A292044: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:   4,     2;
  2:  13,    16,      3;
  3:  32,    84,     40,      4;
  4:  65,   312,    306,     80,      5;
  5: 116,   902,   1584,    836,    140,      6;
  6: 189,  2184,   6149,   5720,   1911,    224,     7;
  7: 288,  4648,  19344,  28600,  16640,   3864,   336,     8;
  8: 417,  8976,  52020, 113152, 103870,  41616,  7140,   480,   9;
  9: 580, 16074, 124032, 374680, 503880, 316540, 93024, 12312, 660, 10;

The row polynomials satisfy (2*n^3-3*n^2+4*n+(4*n^2-7*n)*x+2*n*x^2)*P(n,x) = (2*n^3+3*n^2+4*n+3+(6*n^3-3*n^2-n-8)*x+(12*n^2-17*n+7)*x^2+(6*n-2)*x^3)*P(n-1,x) + ((4*n^2+2)*x-(4*n^2-8*n+8)*x^2-(8*n-10)*x^3-4*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 + 3*n^2 + 4*n + 3 + (6*n^3 - 3*n^2 - n - 8)*x + (12*n^2 - 17*n + 7)*x^2 + (6*n - 2)*x^3)*P[n - 1, x] + ((4*n^2 + 2)*x - (4*n^2 - 8*n + 8)*x^2 - (8*n - 10)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + 4*n + (4*n^2 - 7*n)*x + 2*n*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 + 3*n^2 + 4*n + 3 + (6*n^3 - 3*n^2 - n - 8)*x + (12*n^2 - 17*n + 7)*x^2 + (6*n - 2)*x^3)*v[i - 1] + ((4*n^2 + 2)*x - (4*n^2 - 8*n + 8)*x^2 - (8*n - 10)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + 4*n + (4*n^2 - 7*n)*x + 2*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)

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-ec0a39cf3caa604e
• 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-ec0a39cf3caa604e
• 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-ec0a39cf3caa604e
• 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-ec0a39cf3caa604e

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-ec0a39cf3caa604e

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

  • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

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

    • preserver_image reversal, shift 0 e-75f779345b39f910

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)

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