A126351 → D

OEIS matches ?

No OEIS matches.

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:  0;
  1:  2;
  2:  5,    8;
  3:  9,   38,    24;
  4: 14,  110,   195,     64;
  5: 20,  250,   855,    844,    160;
  6: 27,  490,  2730,   5404,   3325,     384;
  7: 35,  868,  7140,  23604,  30345,   12354,    896;
  8: 44, 1428, 16254,  80724, 177870,  158010,  44135,   2048;
  9: 54, 2220, 33390, 232428, 783300, 1225230, 781515, 153368, 4608;

The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) + ((n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n+1+2*n^2*x)*P(n-1,x) - n*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 2*n^2*x)*P[n - 1, x] - n*x^2*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*v[i - 2] + ((n + 1)*x^2 + (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 2*n^2*x)*v[i - 1] - n*x^2*deriv(v[i - 1], x))/(n)); 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-e022596f9143390c
• 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-e022596f9143390c
• 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-e022596f9143390c
• 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-e022596f9143390c

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-e022596f9143390c

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

  • premise (sequence): Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-0c08584d383e1215 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.