A091044 → ⊙ C(n+2,k+1) → D+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:   2,     1;
  2:   6,     7,      1;
  3:  15,    42,     17,       1;
  4:  31,   198,    174,      34,       1;
  5:  56,   715,   1320,     550,      60,       1;
  6:  92,  2093,   7436,    6149,    1456,      97,      1;
  7: 141,  5236,  32578,   50050,   22750,    3388,    147,     1;
  8: 205, 11628, 117096,  309400,  253708,   71400,   7140,   212,   1;
  9: 286, 23541, 360468, 1527144, 2116296, 1050396, 197676, 13908, 294, 1;

The row polynomials satisfy (4*n^5+8*n^4-7*n^3+n^2+48*n+36+(12*n^4+18*n^3-54*n^2-93*n-18)*x+(12*n^3+28*n^2-13*n-42)*x^2+(4*n^2+14*n+12)*x^3)*P(n,x) = (4*n^5+20*n^4+35*n^3+37*n^2+48*n+36+(40*n^5-58*n^4-13*n^3-35*n^2-96*n-18)*x+(120*n^4-210*n^3-34*n^2+n-42)*x^2+(120*n^3-126*n^2+39*n+12)*x^3+(40*n^2-10*n)*x^4)*P(n-1,x) + ((12*n^4+90*n^3+24*n^2-9*n+54)*x-(60*n^4-114*n^3-266*n^2+149*n+156)*x^2-(180*n^3-320*n^2-238*n+240)*x^3-(180*n^2-210*n-6)*x^4-(60*n-36)*x^5)*d/dx P(n-1,x) + ((24*n^3+4*n^2-10*n+12)*x^2+(24*n^3+76*n^2-34*n-48)*x^3+(72*n^2+48*n-72)*x^4+(72*n+12)*x^5+24*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[((4*n^5 + 20*n^4 + 35*n^3 + 37*n^2 + 48*n + 36 + (40*n^5 - 58*n^4 - 13*n^3 - 35*n^2 - 96*n - 18)*x + (120*n^4 - 210*n^3 - 34*n^2 + n - 42)*x^2 + (120*n^3 - 126*n^2 + 39*n + 12)*x^3 + (40*n^2 - 10*n)*x^4)*P[n - 1, x] + ((12*n^4 + 90*n^3 + 24*n^2 - 9*n + 54)*x - (60*n^4 - 114*n^3 - 266*n^2 + 149*n + 156)*x^2 - (180*n^3 - 320*n^2 - 238*n + 240)*x^3 - (180*n^2 - 210*n - 6)*x^4 - (60*n - 36)*x^5)*D[P[n - 1, x], x] + ((24*n^3 + 4*n^2 - 10*n + 12)*x^2 + (24*n^3 + 76*n^2 - 34*n - 48)*x^3 + (72*n^2 + 48*n - 72)*x^4 + (72*n + 12)*x^5 + 24*x^6)*D[P[n - 1, x], {x, 2}])/(4*n^5 + 8*n^4 - 7*n^3 + n^2 + 48*n + 36 + (12*n^4 + 18*n^3 - 54*n^2 - 93*n - 18)*x + (12*n^3 + 28*n^2 - 13*n - 42)*x^2 + (4*n^2 + 14*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] = ((4*n^5 + 20*n^4 + 35*n^3 + 37*n^2 + 48*n + 36 + (40*n^5 - 58*n^4 - 13*n^3 - 35*n^2 - 96*n - 18)*x + (120*n^4 - 210*n^3 - 34*n^2 + n - 42)*x^2 + (120*n^3 - 126*n^2 + 39*n + 12)*x^3 + (40*n^2 - 10*n)*x^4)*v[i - 1] + ((12*n^4 + 90*n^3 + 24*n^2 - 9*n + 54)*x - (60*n^4 - 114*n^3 - 266*n^2 + 149*n + 156)*x^2 - (180*n^3 - 320*n^2 - 238*n + 240)*x^3 - (180*n^2 - 210*n - 6)*x^4 - (60*n - 36)*x^5)*deriv(v[i - 1], x) + ((24*n^3 + 4*n^2 - 10*n + 12)*x^2 + (24*n^3 + 76*n^2 - 34*n - 48)*x^3 + (72*n^2 + 48*n - 72)*x^4 + (72*n + 12)*x^5 + 24*x^6)*deriv(deriv(v[i - 1], x), x))/(4*n^5 + 8*n^4 - 7*n^3 + n^2 + 48*n + 36 + (12*n^4 + 18*n^3 - 54*n^2 - 93*n - 18)*x + (12*n^3 + 28*n^2 - 13*n - 42)*x^2 + (4*n^2 + 14*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-03362dcf8086f907
• 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-03362dcf8086f907
• 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-03362dcf8086f907

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-03362dcf8086f907

  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