A091044 → ⊙ C(n+2,k+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:   5,     2;
  3:  14,    28,      3;
  4:  30,   168,     90,       4;
  5:  55,   660,    990,     220,       5;
  6:  91,  2002,   6435,    4004,     455,      6;
  7: 140,  5096,  30030,   40040,   12740,    840,     7;
  8: 204, 11424, 111384,  272272,  185640,  34272,  1428,    8;
  9: 285, 23256, 348840, 1410864, 1763580, 697680, 81396, 2280, 9;

The row polynomials satisfy (4*n^4+8*n^3-7*n^2-11*n+6)*P(n,x) = (4*n^4+20*n^3+35*n^2+25*n+6+(40*n^4-70*n^3+17*n^2+7*n+6)*x)*P(n-1,x) + ((12*n^3+90*n^2+24*n+9)*x-(60*n^3-78*n^2-18*n-9)*x^2)*d/dx P(n-1,x) + ((24*n^2+4*n+2)*x^2+(24*n^2+4*n+2)*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[((4*n^4 + 20*n^3 + 35*n^2 + 25*n + 6 + (40*n^4 - 70*n^3 + 17*n^2 + 7*n + 6)*x)*P[n - 1, x] + ((12*n^3 + 90*n^2 + 24*n + 9)*x - (60*n^3 - 78*n^2 - 18*n - 9)*x^2)*D[P[n - 1, x], x] + ((24*n^2 + 4*n + 2)*x^2 + (24*n^2 + 4*n + 2)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^4 + 8*n^3 - 7*n^2 - 11*n + 6)]];
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] = ((4*n^4 + 20*n^3 + 35*n^2 + 25*n + 6 + (40*n^4 - 70*n^3 + 17*n^2 + 7*n + 6)*x)*v[i - 1] + ((12*n^3 + 90*n^2 + 24*n + 9)*x - (60*n^3 - 78*n^2 - 18*n - 9)*x^2)*deriv(v[i - 1], x) + ((24*n^2 + 4*n + 2)*x^2 + (24*n^2 + 4*n + 2)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^4 + 8*n^3 - 7*n^2 - 11*n + 6)); 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-69ea95183e485696
• 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-69ea95183e485696
• 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-69ea95183e485696

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-69ea95183e485696

  Laguerre’s theorem on differential operators [SP14] 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