A125662 → (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,      4;
  2:    8,     24,      9;
  3:   21,    100,     81,     16;
  4:   55,    360,    459,    192,     25;
  5:  144,   1200,   2106,   1376,    375,     36;
  6:  377,   3816,   8559,   7680,   3250,    648,     49;
  7:  987,  11756,  32157,  36880,  21375,   6588,   1029,    64;
  8: 2584,  35400, 114363, 160128, 118500,  49896,  12005,  1536,   81;
  9: 6765, 104780, 390582, 646816, 584325, 313740, 102900, 20224, 2187, 100;

The row polynomials satisfy (200*n+30*n*x-(100*n^3-18*n)*x^2-30*n^3*x^3+2*n^5*x^4)*P(n,x) = (300*n+300+(175*n^2+445*n+270)*x-(150*n^3+165*n^2-87*n-102)*x^2-(10*n^4+130*n^3+205*n^2+79*n-6)*x^3+(3*n^5+3*n^4-27*n^3-54*n^2-33*n-6)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (125+225*x-(175*n^2+175*n-110)*x^2-(285*n^2+285*n+6)*x^3+(5*n^4+10*n^3-120*n^2-125*n-19)*x^4+(6*n^4+12*n^3-9*n^2-15*n-3)*x^5+(n^4+2*n^3+n^2)*x^6)*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[((300*n + 300 + (175*n^2 + 445*n + 270)*x - (150*n^3 + 165*n^2 - 87*n - 102)*x^2 - (10*n^4 + 130*n^3 + 205*n^2 + 79*n - 6)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (125 + 225*x - (175*n^2 + 175*n - 110)*x^2 - (285*n^2 + 285*n + 6)*x^3 + (5*n^4 + 10*n^3 - 120*n^2 - 125*n - 19)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(200*n + 30*n*x - (100*n^3 - 18*n)*x^2 - 30*n^3*x^3 + 2*n^5*x^4)]];
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] = ((300*n + 300 + (175*n^2 + 445*n + 270)*x - (150*n^3 + 165*n^2 - 87*n - 102)*x^2 - (10*n^4 + 130*n^3 + 205*n^2 + 79*n - 6)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (125 + 225*x - (175*n^2 + 175*n - 110)*x^2 - (285*n^2 + 285*n + 6)*x^3 + (5*n^4 + 10*n^3 - 120*n^2 - 125*n - 19)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(200*n + 30*n*x - (100*n^3 - 18*n)*x^2 - 30*n^3*x^3 + 2*n^5*x^4)); 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-0c1d5c17f0bf256d
• 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-0c1d5c17f0bf256d
• 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-0c1d5c17f0bf256d
• 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-0c1d5c17f0bf256d

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-0c1d5c17f0bf256d

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

  • premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-af8f298ca8f41adf root: theorem check

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

    • theorem_check three_term_recurrence e-c7df247a48c8552e 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