A334218 → θ+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: 0,  4;
   4: 0,  6,    9;
   5: 0,  8,   48,     16;
   6: 0, 10,  165,    220,     25;
   7: 0, 12,  468,   1584,    780,     36;
   8: 0, 14, 1197,   8456,  10570,   2394,     49;
   9: 0, 16, 2880,  38112,  96640,  57168,   6720,    64;
  10: 0, 18, 6669, 154548, 702855, 843426, 270459, 17784, 81;

The row polynomials satisfy (n^3-7*n^2+16*n-12)*P(n,x) = ((2*n^4-14*n^3+34*n^2-34*n+12)*x-(n^5-9*n^4+31*n^3-51*n^2+40*n-12)*x^2)*P(n-2,x) + ((n^3-4*n^2+5*n-2)*x-(n^4-5*n^3+9*n^2-7*n+2)*x^2+(n^4-6*n^3+13*n^2-12*n+4)*x^3)*d/dx P(n-2,x) - (n^3-5*n^2+7*n-3-(2*n^4-16*n^3+46*n^2-56*n+24)*x)*P(n-1,x) + ((n^3-6*n^2+11*n-6)*x-(n^3-6*n^2+11*n-6)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 4*x.

P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n^4 - 14*n^3 + 34*n^2 - 34*n + 12)*x - (n^5 - 9*n^4 + 31*n^3 - 51*n^2 + 40*n - 12)*x^2)*P[n - 2, x] + ((n^3 - 4*n^2 + 5*n - 2)*x - (n^4 - 5*n^3 + 9*n^2 - 7*n + 2)*x^2 + (n^4 - 6*n^3 + 13*n^2 - 12*n + 4)*x^3)*D[P[n - 2, x], x] - (n^3 - 5*n^2 + 7*n - 3 - (2*n^4 - 16*n^3 + 46*n^2 - 56*n + 24)*x)*P[n - 1, x] + ((n^3 - 6*n^2 + 11*n - 6)*x - (n^3 - 6*n^2 + 11*n - 6)*x^2)*D[P[n - 1, x], x])/(n^3 - 7*n^2 + 16*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 4*x; for(n = 4, N, my(i = n); v[i] = (((2*n^4 - 14*n^3 + 34*n^2 - 34*n + 12)*x - (n^5 - 9*n^4 + 31*n^3 - 51*n^2 + 40*n - 12)*x^2)*v[i - 2] + ((n^3 - 4*n^2 + 5*n - 2)*x - (n^4 - 5*n^3 + 9*n^2 - 7*n + 2)*x^2 + (n^4 - 6*n^3 + 13*n^2 - 12*n + 4)*x^3)*deriv(v[i - 2], x) - (n^3 - 5*n^2 + 7*n - 3 - (2*n^4 - 16*n^3 + 46*n^2 - 56*n + 24)*x)*v[i - 1] + ((n^3 - 6*n^2 + 11*n - 6)*x - (n^3 - 6*n^2 + 11*n - 6)*x^2)*deriv(v[i - 1], x))/(n^3 - 7*n^2 + 16*n - 12)); 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-ad337510ccdadb92
• 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-ad337510ccdadb92
• 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-ad337510ccdadb92
• 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-ad337510ccdadb92

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-ad337510ccdadb92

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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) → 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.