A059344 → ⊙ 1/k! (shift -1) → D+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:  3,    2;
   4:  7,    6;
   5: 13,   24,    6;
   6: 21,   80,   30;
   7: 31,  210,  150,   20;
   8: 43,  462,  630,  140;
   9: 57,  896, 2100,  840,  70;
  10: 73, 1584, 5796, 4200, 630;

The row polynomials satisfy (n^3-6*n^2+12*n-7+(4*n^2-19*n+15)*x+(4*n-4)*x^2)*P(n,x) = (n^3-4*n^2+6*n-3+(4*n^3-24*n^2+37*n-17)*x+(16*n^2-72*n+84)*x^2+(16*n-32)*x^3)*P(n-1,x) + ((2*n^2-6*n+4)*x-(8*n^2-32*n+30)*x^2-(32*n-64)*x^3-32*x^4)*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^3 - 4*n^2 + 6*n - 3 + (4*n^3 - 24*n^2 + 37*n - 17)*x + (16*n^2 - 72*n + 84)*x^2 + (16*n - 32)*x^3)*P[n - 1, x] + ((2*n^2 - 6*n + 4)*x - (8*n^2 - 32*n + 30)*x^2 - (32*n - 64)*x^3 - 32*x^4)*D[P[n - 1, x], x])/(n^3 - 6*n^2 + 12*n - 7 + (4*n^2 - 19*n + 15)*x + (4*n - 4)*x^2)]];
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^3 - 4*n^2 + 6*n - 3 + (4*n^3 - 24*n^2 + 37*n - 17)*x + (16*n^2 - 72*n + 84)*x^2 + (16*n - 32)*x^3)*v[i - 1] + ((2*n^2 - 6*n + 4)*x - (8*n^2 - 32*n + 30)*x^2 - (32*n - 64)*x^3 - 32*x^4)*deriv(v[i - 1], x))/(n^3 - 6*n^2 + 12*n - 7 + (4*n^2 - 19*n + 15)*x + (4*n - 4)*x^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-272eb8130991447b
• 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-272eb8130991447b
• 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-272eb8130991447b

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-272eb8130991447b

  Laguerre’s theorem on differential operators [SP14] 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) → D+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.