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

OEIS matches ?

• A350265: 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:   5,     6,      1;
  3:  11,    30,     13,      1;
  4:  21,   120,    110,     24,      1;
  5:  36,   385,    700,    315,     40,      1;
  6:  57,  1036,   3430,   2940,    770,     62,     1;
  7:  85,  2436,  13524,  20580,   9996,   1680,    91,    1;
  8: 121,  5160,  44856, 112896,  95256,  29232,  3360,  128,   1;
  9: 166, 10065, 129690, 507276, 698544, 365904, 76230, 6270, 174, 1;

The row polynomials satisfy (n^5+7*n^4+11*n^3-n^2+30*n+72+(3*n^4+18*n^3+9*n^2-78*n-72)*x+(3*n^3+19*n^2+22*n-24)*x^2+(n^2+7*n+12)*x^3)*P(n,x) = (n^5+10*n^4+35*n^3+56*n^2+66*n+72+(10*n^5+13*n^4-4*n^3-n^2-66*n-72)*x+(30*n^4+15*n^3-61*n^2-40*n-24)*x^2+(30*n^3+21*n^2-3*n+12)*x^3+(10*n^2+10*n)*x^4)*P(n-1,x) + ((3*n^4+36*n^3+57*n^2+24*n+72)*x-(15*n^4+3*n^3-113*n^2-32*n+84)*x^2-(45*n^3-8*n^2-155*n+24)*x^3-(45*n^2-3*n-24)*x^4-15*n*x^5)*d/dx P(n-1,x) + ((6*n^3+10*n^2+n+12)*x^2+(6*n^3+28*n^2+13*n-12)*x^3+(18*n^2+30*n-18)*x^4+(18*n+12)*x^5+6*x^6)*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[((n^5 + 10*n^4 + 35*n^3 + 56*n^2 + 66*n + 72 + (10*n^5 + 13*n^4 - 4*n^3 - n^2 - 66*n - 72)*x + (30*n^4 + 15*n^3 - 61*n^2 - 40*n - 24)*x^2 + (30*n^3 + 21*n^2 - 3*n + 12)*x^3 + (10*n^2 + 10*n)*x^4)*P[n - 1, x] + ((3*n^4 + 36*n^3 + 57*n^2 + 24*n + 72)*x - (15*n^4 + 3*n^3 - 113*n^2 - 32*n + 84)*x^2 - (45*n^3 - 8*n^2 - 155*n + 24)*x^3 - (45*n^2 - 3*n - 24)*x^4 - 15*n*x^5)*D[P[n - 1, x], x] + ((6*n^3 + 10*n^2 + n + 12)*x^2 + (6*n^3 + 28*n^2 + 13*n - 12)*x^3 + (18*n^2 + 30*n - 18)*x^4 + (18*n + 12)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 + 7*n^4 + 11*n^3 - n^2 + 30*n + 72 + (3*n^4 + 18*n^3 + 9*n^2 - 78*n - 72)*x + (3*n^3 + 19*n^2 + 22*n - 24)*x^2 + (n^2 + 7*n + 12)*x^3)]];
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^5 + 10*n^4 + 35*n^3 + 56*n^2 + 66*n + 72 + (10*n^5 + 13*n^4 - 4*n^3 - n^2 - 66*n - 72)*x + (30*n^4 + 15*n^3 - 61*n^2 - 40*n - 24)*x^2 + (30*n^3 + 21*n^2 - 3*n + 12)*x^3 + (10*n^2 + 10*n)*x^4)*v[i - 1] + ((3*n^4 + 36*n^3 + 57*n^2 + 24*n + 72)*x - (15*n^4 + 3*n^3 - 113*n^2 - 32*n + 84)*x^2 - (45*n^3 - 8*n^2 - 155*n + 24)*x^3 - (45*n^2 - 3*n - 24)*x^4 - 15*n*x^5)*deriv(v[i - 1], x) + ((6*n^3 + 10*n^2 + n + 12)*x^2 + (6*n^3 + 28*n^2 + 13*n - 12)*x^3 + (18*n^2 + 30*n - 18)*x^4 + (18*n + 12)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 + 7*n^4 + 11*n^3 - n^2 + 30*n + 72 + (3*n^4 + 18*n^3 + 9*n^2 - 78*n - 72)*x + (3*n^3 + 19*n^2 + 22*n - 24)*x^2 + (n^2 + 7*n + 12)*x^3)); 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-5336a01b25274fbd
• 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-5336a01b25274fbd
• 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-5336a01b25274fbd

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-5336a01b25274fbd

  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