A145597 → (θ+1)²

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:  3,   12;
  2:  6,   60,     54;
  3: 10,  180,    405,    160;
  4: 15,  420,   1701,   1680,     375;
  5: 21,  840,   5292,   9408,    5250,     756;
  6: 28, 1512,  13608,  37632,   37800,   13608,    1372;
  7: 36, 2520,  30618, 120960,  189000,  122472,   30870,   2304;
  8: 45, 3960,  62370, 332640,  742500,  748440,  339570,  63360,   3645;
  9: 55, 5940, 117612, 813120, 2450250, 3528360, 2490180, 836352, 120285, 5500;

The row polynomials satisfy (4*n^2+24*n+(5*n^4+41*n^3+60*n^2-36*n)*x+(n^6+12*n^5+45*n^4+54*n^3)*x^2)*P(n,x) = (4*n^2+32*n+48+(5*n^4+51*n^3+190*n^2+244*n+56)*x+(n^6+14*n^5+93*n^4+304*n^3+444*n^2+256*n+64)*x^2+(3*n^6+36*n^5+164*n^4+363*n^3+430*n^2+276*n+72)*x^3)*P(n-1,x) + ((8*n+16)*x+(10*n^3+57*n^2+80*n+12)*x^2+(2*n^5+22*n^4+78*n^3+98*n^2+32*n+8)*x^3-(2*n^5+22*n^4+88*n^3+155*n^2+120*n+36)*x^4)*d/dx 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^2 + 32*n + 48 + (5*n^4 + 51*n^3 + 190*n^2 + 244*n + 56)*x + (n^6 + 14*n^5 + 93*n^4 + 304*n^3 + 444*n^2 + 256*n + 64)*x^2 + (3*n^6 + 36*n^5 + 164*n^4 + 363*n^3 + 430*n^2 + 276*n + 72)*x^3)*P[n - 1, x] + ((8*n + 16)*x + (10*n^3 + 57*n^2 + 80*n + 12)*x^2 + (2*n^5 + 22*n^4 + 78*n^3 + 98*n^2 + 32*n + 8)*x^3 - (2*n^5 + 22*n^4 + 88*n^3 + 155*n^2 + 120*n + 36)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 24*n + (5*n^4 + 41*n^3 + 60*n^2 - 36*n)*x + (n^6 + 12*n^5 + 45*n^4 + 54*n^3)*x^2)]];
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^2 + 32*n + 48 + (5*n^4 + 51*n^3 + 190*n^2 + 244*n + 56)*x + (n^6 + 14*n^5 + 93*n^4 + 304*n^3 + 444*n^2 + 256*n + 64)*x^2 + (3*n^6 + 36*n^5 + 164*n^4 + 363*n^3 + 430*n^2 + 276*n + 72)*x^3)*v[i - 1] + ((8*n + 16)*x + (10*n^3 + 57*n^2 + 80*n + 12)*x^2 + (2*n^5 + 22*n^4 + 78*n^3 + 98*n^2 + 32*n + 8)*x^3 - (2*n^5 + 22*n^4 + 88*n^3 + 155*n^2 + 120*n + 36)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 24*n + (5*n^4 + 41*n^3 + 60*n^2 - 36*n)*x + (n^6 + 12*n^5 + 45*n^4 + 54*n^3)*x^2)); 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-b52fd312b43db576
• 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-b52fd312b43db576
• 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-b52fd312b43db576
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-b52fd312b43db576

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-b52fd312b43db576

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

  • premise (sequence): Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-bcb2e3135d28c083 root: theorem check

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

    • preserver_image reversal, shift 0 e-e8333902099cfe55

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-bcb2e3135d28c083 (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