A039756 → 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:  1;
  2:  4,    2;
  3:  9,   26,     3;
  4: 16,  116,   120,      4;
  5: 25,  340,   990,    484,       5;
  6: 36,  790,  4560,   7084,    1820,       6;
  7: 49, 1582, 15225,  49644,   46095,    6558,       7;
  8: 64, 2856, 41328, 232344,  480480,  283128,   22960,     8;
  9: 81, 4776, 97020, 838488, 3094350, 4319160, 1674540, 78728, 9;

The row polynomials satisfy (2*n-3)*P(n,x) = -(2*n-1+(8*n^2-24*n+10)*x+(8*n^3-36*n^2+46*n-15)*x^2)*P(n-2,x) + ((4*n-2)*x^2+(8*n^2-16*n+6)*x^3)*d/dx P(n-2,x) + (4*n-4+(8*n^2-24*n+18)*x)*P(n-1,x) - (4*n-6)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*P[n - 2, x] + ((4*n - 2)*x^2 + (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], x] + (4*n - 4 + (8*n^2 - 24*n + 18)*x)*P[n - 1, x] - (4*n - 6)*x^2*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 1 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*v[i - 2] + ((4*n - 2)*x^2 + (8*n^2 - 16*n + 6)*x^3)*deriv(v[i - 2], x) + (4*n - 4 + (8*n^2 - 24*n + 18)*x)*v[i - 1] - (4*n - 6)*x^2*deriv(v[i - 1], x))/(2*n - 3)); 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-30b748ef6ad14664
• 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-30b748ef6ad14664
• 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-30b748ef6ad14664
• 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-30b748ef6ad14664

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-30b748ef6ad14664

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

  • premise (sequence): Triangle of B-analogs of Stirling numbers of 2nd kind.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-7519e7535554cb98 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.