A334218 → D+1

OEIS matches ?

• A171005: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 3 ?

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:
   1: 1;
   2: 1;
   3: 2,    2;
   4: 3,    9,      3;
   5: 4,   36,     28,      4;
   6: 5,  115,    220,     75,      5;
   7: 6,  318,   1344,   1020,    186,      6;
   8: 7,  805,   6741,  10570,   4109,    441,     7;
   9: 8, 1928,  29544,  86840,  66968,  15288,  1016,    8;
  10: 9, 4455, 118134, 600921, 843426, 372393, 54198, 2295, 9;

The row polynomials satisfy (n^3-9*n^2+26*n-24+(2*n^2-10*n+12)*x)*P(n,x) = -((n^5-14*n^4+74*n^3-184*n^2+213*n-90)*x^2+(3*n^4-31*n^3+111*n^2-161*n+78)*x^3+(2*n^3-12*n^2+22*n-12)*x^4)*P(n-2,x) - ((n^4-10*n^3+35*n^2-50*n+24)*x^2-(n^4-13*n^3+55*n^2-89*n+46)*x^3-(3*n^3-22*n^2+45*n-26)*x^4-(2*n^2-6*n+4)*x^5)*d/dx P(n-2,x) + (n^3-8*n^2+19*n-12+(2*n^4-23*n^3+96*n^2-165*n+90)*x+(5*n^3-39*n^2+91*n-57)*x^2+(2*n^2-8*n+6)*x^3)*P(n-1,x) + ((n^3-8*n^2+19*n-12)*x-(n^3-10*n^2+27*n-18)*x^2-(2*n^2-8*n+6)*x^3)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + 2*x.

P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^5 - 14*n^4 + 74*n^3 - 184*n^2 + 213*n - 90)*x^2 + (3*n^4 - 31*n^3 + 111*n^2 - 161*n + 78)*x^3 + (2*n^3 - 12*n^2 + 22*n - 12)*x^4)*P[n - 2, x] - ((n^4 - 10*n^3 + 35*n^2 - 50*n + 24)*x^2 - (n^4 - 13*n^3 + 55*n^2 - 89*n + 46)*x^3 - (3*n^3 - 22*n^2 + 45*n - 26)*x^4 - (2*n^2 - 6*n + 4)*x^5)*D[P[n - 2, x], x] + (n^3 - 8*n^2 + 19*n - 12 + (2*n^4 - 23*n^3 + 96*n^2 - 165*n + 90)*x + (5*n^3 - 39*n^2 + 91*n - 57)*x^2 + (2*n^2 - 8*n + 6)*x^3)*P[n - 1, x] + ((n^3 - 8*n^2 + 19*n - 12)*x - (n^3 - 10*n^2 + 27*n - 18)*x^2 - (2*n^2 - 8*n + 6)*x^3)*D[P[n - 1, x], x])/(n^3 - 9*n^2 + 26*n - 24 + (2*n^2 - 10*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 2 + 2*x; for(n = 4, N, my(i = n); v[i] = (-((n^5 - 14*n^4 + 74*n^3 - 184*n^2 + 213*n - 90)*x^2 + (3*n^4 - 31*n^3 + 111*n^2 - 161*n + 78)*x^3 + (2*n^3 - 12*n^2 + 22*n - 12)*x^4)*v[i - 2] - ((n^4 - 10*n^3 + 35*n^2 - 50*n + 24)*x^2 - (n^4 - 13*n^3 + 55*n^2 - 89*n + 46)*x^3 - (3*n^3 - 22*n^2 + 45*n - 26)*x^4 - (2*n^2 - 6*n + 4)*x^5)*deriv(v[i - 2], x) + (n^3 - 8*n^2 + 19*n - 12 + (2*n^4 - 23*n^3 + 96*n^2 - 165*n + 90)*x + (5*n^3 - 39*n^2 + 91*n - 57)*x^2 + (2*n^2 - 8*n + 6)*x^3)*v[i - 1] + ((n^3 - 8*n^2 + 19*n - 12)*x - (n^3 - 10*n^2 + 27*n - 18)*x^2 - (2*n^2 - 8*n + 6)*x^3)*deriv(v[i - 1], x))/(n^3 - 9*n^2 + 26*n - 24 + (2*n^2 - 10*n + 12)*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-63c86030c8ae0f8e
• 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-63c86030c8ae0f8e
• 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-63c86030c8ae0f8e
• 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-63c86030c8ae0f8e

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-63c86030c8ae0f8e

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-9baf8deff42ffdae root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

Same fingerprint (1)

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

• A173018 → reverse (shift +1) → 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.