A062190 → D+1

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:   1;
  1:   7,    6;
  2:  15,   56,    21;
  3:  25,  192,   252,     56;
  4:  37,  468,  1224,    840,     126;
  5:  51,  950,  4050,   5400,    2310,     252;
  6:  67, 1716, 10725,  23100,   18810,    5544,     462;
  7:  85, 2856, 24486,  77000,  100485,   55440,   12012,    792;
  8: 105, 4472, 50232, 216216,  410410,  360360,  144144,  24024,  1287;
  9: 127, 6678, 95004, 535080, 1387386, 1765764, 1117116, 339768, 45045, 2002;

The row polynomials satisfy (n^4+8*n^3+12*n^2-15*n+(2*n^3+11*n^2+5*n)*x+(n^2+5*n)*x^2)*P(n,x) = (n^4+10*n^3+26*n^2+5*n+(3*n^4+26*n^3+68*n^2+65*n)*x+(6*n^3+33*n^2+71*n+40)*x^2+(3*n^2+13*n+20)*x^3)*P(n-1,x) + ((2*n^3+14*n^2+20*n)*x-(2*n^3+10*n^2+4*n-15)*x^2-(4*n^2+14*n+10)*x^3-(2*n+5)*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^4 + 10*n^3 + 26*n^2 + 5*n + (3*n^4 + 26*n^3 + 68*n^2 + 65*n)*x + (6*n^3 + 33*n^2 + 71*n + 40)*x^2 + (3*n^2 + 13*n + 20)*x^3)*P[n - 1, x] + ((2*n^3 + 14*n^2 + 20*n)*x - (2*n^3 + 10*n^2 + 4*n - 15)*x^2 - (4*n^2 + 14*n + 10)*x^3 - (2*n + 5)*x^4)*D[P[n - 1, x], x])/(n^4 + 8*n^3 + 12*n^2 - 15*n + (2*n^3 + 11*n^2 + 5*n)*x + (n^2 + 5*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] = ((n^4 + 10*n^3 + 26*n^2 + 5*n + (3*n^4 + 26*n^3 + 68*n^2 + 65*n)*x + (6*n^3 + 33*n^2 + 71*n + 40)*x^2 + (3*n^2 + 13*n + 20)*x^3)*v[i - 1] + ((2*n^3 + 14*n^2 + 20*n)*x - (2*n^3 + 10*n^2 + 4*n - 15)*x^2 - (4*n^2 + 14*n + 10)*x^3 - (2*n + 5)*x^4)*deriv(v[i - 1], x))/(n^4 + 8*n^3 + 12*n^2 - 15*n + (2*n^3 + 11*n^2 + 5*n)*x + (n^2 + 5*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-f9926095268ea913
• 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-f9926095268ea913
• 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-f9926095268ea913
• 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-f9926095268ea913

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-f9926095268ea913

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

  • premise (sequence): Coefficient triangle of certain polynomials N(5; m,x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-e968c429d4b85e0b 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.