A108838 → ⊙ C(n+1,k+1) (shift +1) → x ↦ x+1

OEIS matches ?

• A368708: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:      2,       1;
  2:      6,       6,       1;
  3:     22,      33,      13,       1;
  4:     92,     184,     116,      24,       1;
  5:    422,    1055,     920,     325,      40,      1;
  6:   2074,    6222,    6945,    3520,     785,     62,     1;
  7:  10754,   37639,   51261,   34055,   11291,   1701,    91,    1;
  8:  58202,  232808,  374500,  308672,  137326,  31808,  3388,  128,   1;
  9: 326240, 1468080, 2724780, 2685690, 1507170, 479220, 80934, 6306, 174, 1;

The row polynomials satisfy (n^2+7*n+12)*P(n,x) = (11*n^2+17*n+12+(10*n^2+10*n)*x)*P(n-1,x) - (12*n-24+(27*n-24)*x+15*n*x^2)*d/dx P(n-1,x) + (12+30*x+24*x^2+6*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((11*n^2 + 17*n + 12 + (10*n^2 + 10*n)*x)*P[n - 1, x] - (12*n - 24 + (27*n - 24)*x + 15*n*x^2)*D[P[n - 1, x], x] + (12 + 30*x + 24*x^2 + 6*x^3)*D[P[n - 1, x], {x, 2}])/(n^2 + 7*n + 12)]];
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] = ((11*n^2 + 17*n + 12 + (10*n^2 + 10*n)*x)*v[i - 1] - (12*n - 24 + (27*n - 24)*x + 15*n*x^2)*deriv(v[i - 1], x) + (12 + 30*x + 24*x^2 + 6*x^3)*deriv(deriv(v[i - 1], x), x))/(n^2 + 7*n + 12)); 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, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-0c759266bd791ca5
• 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-0c759266bd791ca5
• 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-0c759266bd791ca5

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 affine, shift 0 e-0c759266bd791ca5

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image hadamard_multiplier, shift 1 e-548c39a476f786f5

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

      • premise (sequence): Triangle of Dyck paths counted by number of long interior inclines.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-a4549745e8cd4154 root: theorem check

          Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • preserver_image reversal, shift 0 e-97ff2ea1491caf9e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ preserver_image hadamard_multiplier, shift 1 e-548c39a476f786f5 (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A108838 → ⊙ C(n+1,k+1) (shift +1) → x ↦ x/(x+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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023