A207823 → reverse

OEIS matches ?

• A207823: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A004254: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1,  4;
  2: 1,  8,  15;
  3: 1, 12,  46,   56;
  4: 1, 16,  93,  232,   209;
  5: 1, 20, 156,  592,  1091,    780;
  6: 1, 24, 235, 1200,  3366,   4912,   2911;
  7: 1, 28, 330, 2120,  8010,  17784,  21468,  10864;
  8: 1, 32, 441, 3416, 16255,  48624,  89238,  91824,  40545;
  9: 1, 36, 568, 5152, 29589, 111524, 275724, 430992, 386373, 151316;

The row polynomials satisfy 2*n*P(n,x) = (2*n+(12*n-4)*x+(12*n-12)*x^2)*P(n-1,x) - (x+8*x^2+12*x^3)*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[((2*n + (12*n - 4)*x + (12*n - 12)*x^2)*P[n - 1, x] - (x + 8*x^2 + 12*x^3)*D[P[n - 1, x], x])/(2*n)]];
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] = ((2*n + (12*n - 4)*x + (12*n - 12)*x^2)*v[i - 1] - (x + 8*x^2 + 12*x^3)*deriv(v[i - 1], x))/(2*n)); 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)

Triangle begins:
  0:      1;
  1:      4,      1;
  2:     15,      8,      1;
  3:     56,     46,     12,      1;
  4:    209,    232,     93,     16,      1;
  5:    780,   1091,    592,    156,     20,     1;
  6:   2911,   4912,   3366,   1200,    235,    24,    1;
  7:  10864,  21468,  17784,   8010,   2120,   330,   28,   1;
  8:  40545,  91824,  89238,  48624,  16255,  3416,  441,  32,  1;
  9: 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where 2*n*P(n,x) = (2*n+(12*n-4)*x+(12*n-12)*x^2)*P(n-1,x) - (x+8*x^2+12*x^3)*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[((2*n + (12*n - 4)*x + (12*n - 12)*x^2)*P[n - 1, x] - (x + 8*x^2 + 12*x^3)*D[P[n - 1, x], x])/(2*n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + (12*n - 4)*x + (12*n - 12)*x^2)*v[i - 1] - (x + 8*x^2 + 12*x^3)*deriv(v[i - 1], x))/(2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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 (-1/2, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-73dd335e950d1f81
• 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-73dd335e950d1f81
• 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-73dd335e950d1f81

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 reversal, shift 0 e-73dd335e950d1f81

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, -2)
    • preserver_image affine, shift 0 e-56c0c02612a8191c

      Affine, reversal and Möbius substitutions (elementary) 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 (-5, -1)
        • theorem_check three_term_recurrence e-c7df247a48c8552e root: theorem check

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

    • theorem_check three_term_recurrence e-ec629e395a3def0c root: theorem check

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

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023