A001263 → D+1

OEIS matches ?

• A001700: 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 ?
• A088218: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  7,   18,    9,     1;
  4: 11,   50,   50,    14,     1;
  5: 16,  115,  200,   110,    20,     1;
  6: 22,  231,  630,   595,   210,    27,    1;
  7: 29,  420, 1666,  2450,  1470,   364,   35,   1;
  8: 37,  708, 3864,  8232,  7644,  3192,  588,  44,  1;
  9: 46, 1125, 8100, 23688, 31752, 20412, 6300, 900, 54, 1;

The row polynomials satisfy (n^3+n^2+4+(2*n^2+n-6)*x+(n+2)*x^2)*P(n,x) = (n^3+3*n^2+4*n+4+(3*n^3+n^2+2*n-6)*x+(6*n^2-5*n+2)*x^2+3*n*x^3)*P(n-1,x) + ((2*n^2+n+2)*x-(2*n^2-3*n+4)*x^2-(4*n-4)*x^3-2*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[((n^3 + 3*n^2 + 4*n + 4 + (3*n^3 + n^2 + 2*n - 6)*x + (6*n^2 - 5*n + 2)*x^2 + 3*n*x^3)*P[n - 1, x] + ((2*n^2 + n + 2)*x - (2*n^2 - 3*n + 4)*x^2 - (4*n - 4)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 + 4 + (2*n^2 + n - 6)*x + (n + 2)*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] = ((n^3 + 3*n^2 + 4*n + 4 + (3*n^3 + n^2 + 2*n - 6)*x + (6*n^2 - 5*n + 2)*x^2 + 3*n*x^3)*v[i - 1] + ((2*n^2 + n + 2)*x - (2*n^2 - 3*n + 4)*x^2 - (4*n - 4)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 + 4 + (2*n^2 + n - 6)*x + (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 - 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-bb9e94713e97c63a
• 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-bb9e94713e97c63a
• 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-bb9e94713e97c63a
• 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-bb9e94713e97c63a

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-bb9e94713e97c63a

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

  • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-d285019b29e04549 root: theorem check

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

    • preserver_image reversal, shift 0 e-14da04f2a2dce119

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-d285019b29e04549 (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• narayana → D+1

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