A104684 → (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:
   1:     1;
   2:     2,      4;
   3:     6,     24,       9;
   4:    20,    120,     108,      16;
   5:    70,    560,     810,     320,      25;
   6:   252,   2520,    5040,    3360,     750,      36;
   7:   924,  11088,   28350,   26880,   10500,    1512,     49;
   8:  3432,  48048,  149688,  184800,  105000,   27216,   2744,     64;
   9: 12870, 205920,  756756, 1153152,  866250,  332640,  61740,   4608,   81;
  10: 48620, 875160, 3706560, 6726720, 6306300, 3243240, 905520, 126720, 7290, 100;

The row polynomials satisfy (4*n^2-8*n+4+(6*n^3-21*n^2+24*n-9)*x+(n^4-4*n^3+6*n^2-4*n+1)*x^2)*P(n,x) = (16*n^2-40*n+24+(24*n^3-48*n^2-2*n+26)*x+(4*n^4+12*n^3-58*n^2+42*n+2)*x^2+(3*n^4-6*n^3-n^2+4*n)*x^3)*P(n-1,x) - ((8*n-8)*x+(12*n^2-4*n-9)*x^2+(2*n^3+12*n^2-14*n-1)*x^3+(2*n^3-2*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 40*n + 24 + (24*n^3 - 48*n^2 - 2*n + 26)*x + (4*n^4 + 12*n^3 - 58*n^2 + 42*n + 2)*x^2 + (3*n^4 - 6*n^3 - n^2 + 4*n)*x^3)*P[n - 1, x] - ((8*n - 8)*x + (12*n^2 - 4*n - 9)*x^2 + (2*n^3 + 12*n^2 - 14*n - 1)*x^3 + (2*n^3 - 2*n)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 8*n + 4 + (6*n^3 - 21*n^2 + 24*n - 9)*x + (n^4 - 4*n^3 + 6*n^2 - 4*n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^2 - 40*n + 24 + (24*n^3 - 48*n^2 - 2*n + 26)*x + (4*n^4 + 12*n^3 - 58*n^2 + 42*n + 2)*x^2 + (3*n^4 - 6*n^3 - n^2 + 4*n)*x^3)*v[i - 1] - ((8*n - 8)*x + (12*n^2 - 4*n - 9)*x^2 + (2*n^3 + 12*n^2 - 14*n - 1)*x^3 + (2*n^3 - 2*n)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 8*n + 4 + (6*n^3 - 21*n^2 + 24*n - 9)*x + (n^4 - 4*n^3 + 6*n^2 - 4*n + 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-8470b587e8da7dfc
• Coefficients of one sign. Every nonzero row, n >= 1: 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-8470b587e8da7dfc
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-8470b587e8da7dfc
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-8470b587e8da7dfc

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-8470b587e8da7dfc

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-74c20febd3423e2e root: theorem check

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

    • preserver_image reversal, shift 0 e-5047c71cb920acb6

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-18de01688f1c88b0 root: theorem check

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

        • theorem_check three_term_recurrence e-93f30859d3148f4f root: theorem check

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

        • preserver_image reversal, shift 0 e-00d47151fcd2720e

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-74c20febd3423e2e (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