A173018 → D+1

OEIS matches ?

• A001710: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A357711: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A188588: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:
   1:   1;
   2:   1;
   3:   2,     1;
   4:   5,     6,      1;
   5:  12,    33,     14,      1;
   6:  27,   158,    144,     30,      1;
   7:  58,   661,   1208,    530,     62,      1;
   8: 121,  2502,   8439,   7180,   1791,    126,     1;
   9: 248,  8833,  51150,  78095,  37084,   5775,   254,   1;
  10: 503, 29718, 279310, 712994, 597360, 175882, 18122, 510, 1;

The row polynomials satisfy (n-4+2*x)*P(n,x) = -(2*n-6+(2*n^2-15*n+31)*x+(n^3-11*n^2+44*n-66)*x^2+(3*n^2-22*n+41)*x^3+(2*n-6)*x^4)*P(n-2,x) - ((n-3)*x+(n^2-8*n+17)*x^2-(n^2-10*n+25)*x^3-(3*n-13)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-11+(2*n^2-15*n+34)*x+(5*n-19)*x^2+2*x^3)*P(n-1,x) + ((n-4)*x-(n-6)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 6 + (2*n^2 - 15*n + 31)*x + (n^3 - 11*n^2 + 44*n - 66)*x^2 + (3*n^2 - 22*n + 41)*x^3 + (2*n - 6)*x^4)*P[n - 2, x] - ((n - 3)*x + (n^2 - 8*n + 17)*x^2 - (n^2 - 10*n + 25)*x^3 - (3*n - 13)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 11 + (2*n^2 - 15*n + 34)*x + (5*n - 19)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 4)*x - (n - 6)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 4 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n - 6 + (2*n^2 - 15*n + 31)*x + (n^3 - 11*n^2 + 44*n - 66)*x^2 + (3*n^2 - 22*n + 41)*x^3 + (2*n - 6)*x^4)*v[i - 2] - ((n - 3)*x + (n^2 - 8*n + 17)*x^2 - (n^2 - 10*n + 25)*x^3 - (3*n - 13)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 11 + (2*n^2 - 15*n + 34)*x + (5*n - 19)*x^2 + 2*x^3)*v[i - 1] + ((n - 4)*x - (n - 6)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 4 + 2*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-0053277d0044178d
• Coefficients of one sign. Every nonzero row, n >= 1: 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-0053277d0044178d
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-0053277d0044178d
• 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-0053277d0044178d

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-0053277d0044178d

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

  • premise (sequence): Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-083b494dcfd2f637 root: theorem check

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

    • parent_provers root_domain e-ffc8a9fffd6e96cb root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A008292 → D+1

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.