A059344 → ⊙ 1/k! (shift -1) → θ+1

OEIS matches ?

• A391990: 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,   4;
   4: 1,  12;
   5: 1,  24,   18;
   6: 1,  40,   90;
   7: 1,  60,  270,   80;
   8: 1,  84,  630,  560;
   9: 1, 112, 1260, 2240,  350;
  10: 1, 144, 2268, 6720, 3150;

The row polynomials satisfy -(n-1-(n^3-2*n^2+n)*x)*P(n,x) = -(n-1-(n^3-2*n^2-7*n+16)*x-(4*n^3-4*n^2-8*n)*x^2)*P(n-1,x) - (2*x-(2*n^2+2*n+8)*x^2+(8*n^2+8*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 - (n^3 - 2*n^2 - 7*n + 16)*x - (4*n^3 - 4*n^2 - 8*n)*x^2)*P[n - 1, x] - (2*x - (2*n^2 + 2*n + 8)*x^2 + (8*n^2 + 8*n)*x^3)*D[P[n - 1, x], x])/(-n + 1 + (n^3 - 2*n^2 + n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - 1 - (n^3 - 2*n^2 - 7*n + 16)*x - (4*n^3 - 4*n^2 - 8*n)*x^2)*v[i - 1] - (2*x - (2*n^2 + 2*n + 8)*x^2 + (8*n^2 + 8*n)*x^3)*deriv(v[i - 1], x))/(-n + 1 + (n^3 - 2*n^2 + n)*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-65103a54d767b0ce
• 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-65103a54d767b0ce
• 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-65103a54d767b0ce

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-65103a54d767b0ce

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

  • premise (sequence): T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image hadamard_multiplier, shift -1 e-4d3032ecd7423900

      Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

      • premise (sequence): Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers paper_theorem e-94e95c2ea0369f05 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.

• A011973 → ⊙ C(n,k) → 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.