A367270 → D+1

OEIS matches ?

• A141222: 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:  3,    2;
  3:  7,   12,     3;
  4: 13,   48,    30,     4;
  5: 21,  140,   180,    60,     5;
  6: 31,  330,   750,   500,   105,     6;
  7: 43,  672,  2415,  2800,  1155,   168,    7;
  8: 57, 1232,  6468, 11760,  8330,  2352,  252,   8;
  9: 73, 2088, 15120, 39984, 44100, 21168, 4368, 360, 9;

The row polynomials satisfy (n^4-4*n^3+6*n^2-3*n+(2*n^3-7*n^2+5*n)*x+(n^2-n)*x^2)*P(n,x) = (n^4-2*n^3+2*n^2-n+(3*n^4-10*n^3+8*n^2-n)*x+(6*n^3-21*n^2+23*n-8)*x^2+(3*n^2-5*n+2)*x^3)*P(n-1,x) + ((2*n^3-4*n^2+2*n)*x-(2*n^3-8*n^2+10*n-3)*x^2-(4*n^2-10*n+4)*x^3-(2*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 - 2*n^3 + 2*n^2 - n + (3*n^4 - 10*n^3 + 8*n^2 - n)*x + (6*n^3 - 21*n^2 + 23*n - 8)*x^2 + (3*n^2 - 5*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 - 4*n^2 + 2*n)*x - (2*n^3 - 8*n^2 + 10*n - 3)*x^2 - (4*n^2 - 10*n + 4)*x^3 - (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^3 + 6*n^2 - 3*n + (2*n^3 - 7*n^2 + 5*n)*x + (n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 - 2*n^3 + 2*n^2 - n + (3*n^4 - 10*n^3 + 8*n^2 - n)*x + (6*n^3 - 21*n^2 + 23*n - 8)*x^2 + (3*n^2 - 5*n + 2)*x^3)*v[i - 1] + ((2*n^3 - 4*n^2 + 2*n)*x - (2*n^3 - 8*n^2 + 10*n - 3)*x^2 - (4*n^2 - 10*n + 4)*x^3 - (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^3 + 6*n^2 - 3*n + (2*n^3 - 7*n^2 + 5*n)*x + (n^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-9d22aaa3a73f8d2e
• 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-9d22aaa3a73f8d2e
• 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-9d22aaa3a73f8d2e
• 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-9d22aaa3a73f8d2e

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-9d22aaa3a73f8d2e

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

  • premise (sequence): Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-2b43df6c63e34146 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