A106800 → θ+1

OEIS matches ?

• A359109: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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: 1,  2;
   4: 1,  6,    3;
   5: 1, 12,   21,     4;
   6: 1, 20,   75,    60,     5;
   7: 1, 30,  195,   360,   155,     6;
   8: 1, 42,  420,  1400,  1505,   378,     7;
   9: 1, 56,  798,  4200,  8505,  5796,   889,    8;
  10: 1, 72, 1386, 10584, 34755, 46620, 21175, 2040, 9;

The row polynomials satisfy (n-2)*P(n,x) = -(n-1+(2*n^2-8*n+6)*x+(n^3-6*n^2+11*n-6)*x^2)*P(n-2,x) + ((n-1)*x^2+(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (2*n-3+(2*n^2-8*n+8)*x)*P(n-1,x) - (n-2)*x^2*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[(-(n - 1 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*P[n - 2, x] + ((n - 1)*x^2 + (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n - 3 + (2*n^2 - 8*n + 8)*x)*P[n - 1, x] - (n - 2)*x^2*D[P[n - 1, x], x])/(n - 2)]];
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] = (-(n - 1 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*v[i - 2] + ((n - 1)*x^2 + (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n - 3 + (2*n^2 - 8*n + 8)*x)*v[i - 1] - (n - 2)*x^2*deriv(v[i - 1], x))/(n - 2)); 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-7e1969ff0612cc65
• 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-7e1969ff0612cc65
• 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-7e1969ff0612cc65
• 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-7e1969ff0612cc65

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 euler_polynomial, shift 0 e-7e1969ff0612cc65

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-2ab1dfe057e84c9d root: theorem check

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

Related sequences ?

Same fingerprint (1)

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

• A008278 → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.