A103633 → 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: 1;
  1: 0, 2;
  2: 0, 2, 3;
  3: 0, 0, 3, 4;
  4: 0, 0, 3, 8,  5;
  5: 0, 0, 0, 4, 10,  6;
  6: 0, 0, 0, 4, 15, 18,  7;
  7: 0, 0, 0, 0,  5, 18, 21,  8;
  8: 0, 0, 0, 0,  5, 24, 42, 32,  9;
  9: 0, 0, 0, 0,  0,  6, 28, 48, 36, 10;

The row polynomials satisfy (n^2+3*n+2+(4*n^2+2*n)*x+(4*n^2-4*n)*x^2)*P(n,x) = -(n^3+2*n^2-n-2+(6*n^3+7*n^2-13*n-12)*x+(12*n^3-30*n+6)*x^2+(8*n^3-12*n^2-4*n+8)*x^3)*P(n-1,x) + ((2*n^2+6*n+4)*x+(10*n^2+18*n)*x^2+(16*n^2+12*n-8)*x^3+(8*n^2-4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 2*n^2 - n - 2 + (6*n^3 + 7*n^2 - 13*n - 12)*x + (12*n^3 - 30*n + 6)*x^2 + (8*n^3 - 12*n^2 - 4*n + 8)*x^3)*P[n - 1, x] + ((2*n^2 + 6*n + 4)*x + (10*n^2 + 18*n)*x^2 + (16*n^2 + 12*n - 8)*x^3 + (8*n^2 - 4)*x^4)*D[P[n - 1, x], x])/(n^2 + 3*n + 2 + (4*n^2 + 2*n)*x + (4*n^2 - 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 + 2*n^2 - n - 2 + (6*n^3 + 7*n^2 - 13*n - 12)*x + (12*n^3 - 30*n + 6)*x^2 + (8*n^3 - 12*n^2 - 4*n + 8)*x^3)*v[i - 1] + ((2*n^2 + 6*n + 4)*x + (10*n^2 + 18*n)*x^2 + (16*n^2 + 12*n - 8)*x^3 + (8*n^2 - 4)*x^4)*deriv(v[i - 1], x))/(n^2 + 3*n + 2 + (4*n^2 + 2*n)*x + (4*n^2 - 4*n)*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-1c588a07b00ef8dc
• 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-1c588a07b00ef8dc
• 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-1c588a07b00ef8dc

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-1c588a07b00ef8dc

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

  • premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-35a33ce2a6b89415 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.