A102547 → reverse → θ+1

OEIS matches ?

No OEIS matches.

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:
  0: 0, 2;
  1: 0, 0, 3;
  2: 0, 0, 0, 4;
  3: 0, 0, 0, 4,  5;
  4: 0, 0, 0, 0, 10,  6;
  5: 0, 0, 0, 0,  0, 18,  7;
  6: 0, 0, 0, 0,  0,  6, 28,  8;
  7: 0, 0, 0, 0,  0,  0, 21, 40,  9;
  8: 0, 0, 0, 0,  0,  0,  0, 48, 54, 10;
  9: 0, 0, 0, 0,  0,  0,  0,  8, 90, 70, 11;

The row polynomials satisfy 2*P(n,x) = 12*x^3*P(n-5,x) + 4*x^4*d/dx P(n-5,x) + 2*x^3*P(n-4,x) + x^4*d/dx P(n-4,x) + 6*x^2*P(n-3,x) + 4*x^3*d/dx P(n-3,x) + 4*x*P(n-2,x) - (4*x^2-x^3)*d/dx P(n-2,x) + 4*x*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 5, with P(0,x) = 2*x, P(1,x) = 3*x^2, P(2,x) = 4*x^3, P(3,x) = 4*x^3 + 5*x^4, P(4,x) = 10*x^4 + 6*x^5.

P[0, x_] := 2*x;
P[1, x_] := 3*x^2;
P[2, x_] := 4*x^3;
P[3, x_] := 4*x^3 + 5*x^4;
P[4, x_] := 10*x^4 + 6*x^5;
P[n_, x_] := P[n, x] = Expand[Together[(12*x^3*P[n - 5, x] + 4*x^4*D[P[n - 5, x], x] + 2*x^3*P[n - 4, x] + x^4*D[P[n - 4, x], x] + 6*x^2*P[n - 3, x] + 4*x^3*D[P[n - 3, x], x] + 4*x*P[n - 2, x] - (4*x^2 - x^3)*D[P[n - 2, x], x] + 4*x*P[n - 1, x] - x^2*D[P[n - 1, x], x])/(2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2*x; v[2] = 3*x^2; v[3] = 4*x^3; v[4] = 4*x^3 + 5*x^4; v[5] = 10*x^4 + 6*x^5; for(n = 5, N, my(i = n + 1); v[i] = (12*x^3*v[i - 5] + 4*x^4*deriv(v[i - 5], x) + 2*x^3*v[i - 4] + x^4*deriv(v[i - 4], x) + 6*x^2*v[i - 3] + 4*x^3*deriv(v[i - 3], x) + 4*x*v[i - 2] - (4*x^2 - x^3)*deriv(v[i - 2], x) + 4*x*v[i - 1] - x^2*deriv(v[i - 1], x))/(2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, 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-c5a44f97a1ce25c1
• Coefficients of one sign. Every nonzero row, n >= 0: 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-c5a44f97a1ce25c1
• Hadamard kernel (Malo). Rows P_n, n >= 0, 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-c5a44f97a1ce25c1

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-c5a44f97a1ce25c1

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

  • premise (sequence): Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-0e9e334768ea0a35

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • preserver_image reversal, shift 0 e-9cfbf7a52f4ba322

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            no recorded property covers this requirement
        • parent_provers paper_theorem e-eef18680cd27a7c9 root: theorem check

          Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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.