A111125 → θ+1

OEIS matches ?

• A377857: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  3,   2;
   3:  5,  10,    3;
   4:  7,  28,   21,     4;
   5:  9,  60,   81,    36,     5;
   6: 11, 110,  231,   176,    55,     6;
   7: 13, 182,  546,   624,   325,    78,    7;
   8: 15, 280, 1134,  1800,  1375,   540,  105,    8;
   9: 17, 408, 2142,  4488,  4675,  2652,  833,  136,   9;
  10: 19, 570, 3762, 10032, 13585, 10374, 4655, 1216, 171, 10;

The row polynomials satisfy -(4*n-6-(2*n^3-9*n^2+13*n-6)*x)*P(n,x) = -(4*n-2-(2*n^3-7*n^2+3*n+2)*x-(n^3-3*n^2+2*n)*x^2)*P(n-1,x) - (8*x-(4*n^2-8*n-2)*x^2-(n^2-2*n)*x^3)*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[(-(4*n - 2 - (2*n^3 - 7*n^2 + 3*n + 2)*x - (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 1, x] - (8*x - (4*n^2 - 8*n - 2)*x^2 - (n^2 - 2*n)*x^3)*D[P[n - 1, x], x])/(-4*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*x)]];
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] = (-(4*n - 2 - (2*n^3 - 7*n^2 + 3*n + 2)*x - (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 1] - (8*x - (4*n^2 - 8*n - 2)*x^2 - (n^2 - 2*n)*x^3)*deriv(v[i - 1], x))/(-4*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*x)); 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-1de9fee97176e47c
• 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-1de9fee97176e47c
• 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-1de9fee97176e47c
• 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-1de9fee97176e47c

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-1de9fee97176e47c

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

  • premise (sequence): Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-0bb8ed02ac535b32 root: theorem check

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

    • theorem_check three_term_recurrence e-19ca7a5e6e7a3cfc root: theorem check

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

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