A108838 → ⊙ C(n+1,k+1) (shift +1) → D

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;
  1:   1;
  2:   4,    2;
  3:  10,   20,      3;
  4:  20,  100,     60,      4;
  5:  35,  350,    525,    140,      5;
  6:  56,  980,   2940,   1960,    280,      6;
  7:  84, 2352,  12348,  16464,   5880,    504,     7;
  8: 120, 5040,  42336,  98784,  70560,  15120,   840,    8;
  9: 165, 9900, 124740, 465696, 582120, 249480, 34650, 1320, 9;

The row polynomials satisfy (n^3+6*n^2+5*n-12)*P(n,x) = (n^3+9*n^2+26*n+24+(10*n^3-10*n)*x)*P(n-1,x) + ((3*n^2+33*n+24)*x-(15*n^2-3*n-12)*x^2)*d/dx P(n-1,x) + ((6*n+4)*x^2+(6*n+4)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 9*n^2 + 26*n + 24 + (10*n^3 - 10*n)*x)*P[n - 1, x] + ((3*n^2 + 33*n + 24)*x - (15*n^2 - 3*n - 12)*x^2)*D[P[n - 1, x], x] + ((6*n + 4)*x^2 + (6*n + 4)*x^3)*D[P[n - 1, x], {x, 2}])/(n^3 + 6*n^2 + 5*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 9*n^2 + 26*n + 24 + (10*n^3 - 10*n)*x)*v[i - 1] + ((3*n^2 + 33*n + 24)*x - (15*n^2 - 3*n - 12)*x^2)*deriv(v[i - 1], x) + ((6*n + 4)*x^2 + (6*n + 4)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^3 + 6*n^2 + 5*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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-d09a0797aedc5af0
• 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-d09a0797aedc5af0
• 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-d09a0797aedc5af0

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-d09a0797aedc5af0

  Laguerre’s theorem on differential operators [SP14] 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 ?

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. [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