A091044 → ⊙ C(n+2,k+1) → 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: 1,   2;
  2: 1,  10,     3;
  3: 1,  28,    42,      4;
  4: 1,  60,   252,    120,       5;
  5: 1, 110,   990,   1320,     275,       6;
  6: 1, 182,  3003,   8580,    5005,     546,      7;
  7: 1, 280,  7644,  40040,   50050,   15288,    980,     8;
  8: 1, 408, 17136, 148512,  340340,  222768,  39984,  1632,    9;
  9: 1, 570, 34884, 465120, 1763580, 2116296, 813960, 93024, 2565, 10;

The row polynomials satisfy (4*n^4+12*n^3+11*n^2+3*n)*P(n,x) = (4*n^4+12*n^3+11*n^2+3*n+(40*n^4+30*n^3-7*n^2-3*n)*x)*P(n-1,x) + ((12*n^3+54*n^2+48*n+9)*x-(60*n^3+30*n^2-30*n-9)*x^2)*d/dx P(n-1,x) + ((24*n^2+28*n+6)*x^2+(24*n^2+28*n+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[((4*n^4 + 12*n^3 + 11*n^2 + 3*n + (40*n^4 + 30*n^3 - 7*n^2 - 3*n)*x)*P[n - 1, x] + ((12*n^3 + 54*n^2 + 48*n + 9)*x - (60*n^3 + 30*n^2 - 30*n - 9)*x^2)*D[P[n - 1, x], x] + ((24*n^2 + 28*n + 6)*x^2 + (24*n^2 + 28*n + 6)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^4 + 12*n^3 + 11*n^2 + 3*n)]];
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] = ((4*n^4 + 12*n^3 + 11*n^2 + 3*n + (40*n^4 + 30*n^3 - 7*n^2 - 3*n)*x)*v[i - 1] + ((12*n^3 + 54*n^2 + 48*n + 9)*x - (60*n^3 + 30*n^2 - 30*n - 9)*x^2)*deriv(v[i - 1], x) + ((24*n^2 + 28*n + 6)*x^2 + (24*n^2 + 28*n + 6)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^4 + 12*n^3 + 11*n^2 + 3*n)); 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-b04192f4d27a9c29
• 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-b04192f4d27a9c29
• 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-b04192f4d27a9c29

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-b04192f4d27a9c29

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

  • premise (sequence): Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image hadamard_multiplier, shift 0 e-00224510dcd0cfb5

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

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

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

        • preserver_image reversal, shift 0 e-75f779345b39f910

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)
    • preserver_image reversal, shift 0 e-3a2967256f8a0d67

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ preserver_image hadamard_multiplier, shift 0 e-00224510dcd0cfb5 (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