A207607 → reverse

OEIS matches ?

• A237268: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A001906: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A088305: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A207607: reversed triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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: 2,  1;
  2: 2,  5,   1;
  3: 2,  9,   9,   1;
  4: 2, 13,  25,  14,    1;
  5: 2, 17,  49,  55,   20,    1;
  6: 2, 21,  81, 140,  105,   27,    1;
  7: 2, 25, 121, 285,  336,  182,   35,   1;
  8: 2, 29, 169, 506,  825,  714,  294,  44,  1;
  9: 2, 33, 225, 819, 1716, 2079, 1386, 450, 54, 1;

The row polynomials satisfy (8*n^2-16*n+8)*P(n,x) = (8*n^2-16*n+8+(24*n^2-54*n+30)*x+(12*n^2-30*n+18)*x^2)*P(n-1,x) - ((4*n-4)*x+(22*n-22)*x^2+(24*n-42)*x^3)*d/dx P(n-1,x) + (3*x^3+12*x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 16*n + 8 + (24*n^2 - 54*n + 30)*x + (12*n^2 - 30*n + 18)*x^2)*P[n - 1, x] - ((4*n - 4)*x + (22*n - 22)*x^2 + (24*n - 42)*x^3)*D[P[n - 1, x], x] + (3*x^3 + 12*x^4)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 16*n + 8)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 16*n + 8 + (24*n^2 - 54*n + 30)*x + (12*n^2 - 30*n + 18)*x^2)*v[i - 1] - ((4*n - 4)*x + (22*n - 22)*x^2 + (24*n - 42)*x^3)*deriv(v[i - 1], x) + (3*x^3 + 12*x^4)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 16*n + 8)); 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:
   1: 1;
   2: 1,  2;
   3: 1,  5,   2;
   4: 1,  9,   9,    2;
   5: 1, 14,  25,   13,    2;
   6: 1, 20,  55,   49,   17,    2;
   7: 1, 27, 105,  140,   81,   21,   2;
   8: 1, 35, 182,  336,  285,  121,  25,   2;
   9: 1, 44, 294,  714,  825,  506, 169,  29,  2;
  10: 1, 54, 450, 1386, 2079, 1716, 819, 225, 33, 2;

Row n lists the coefficients of P(n,x) in descending powers of x, where (8*n^2-32*n+32)*P(n,x) = (8*n^2-32*n+32+(24*n^2-102*n+108)*x+(12*n^2-54*n+60)*x^2)*P(n-1,x) - ((4*n-8)*x+(22*n-44)*x^2+(24*n-66)*x^3)*d/dx P(n-1,x) + (3*x^3+12*x^4)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 32*n + 32 + (24*n^2 - 102*n + 108)*x + (12*n^2 - 54*n + 60)*x^2)*P[n - 1, x] - ((4*n - 8)*x + (22*n - 44)*x^2 + (24*n - 66)*x^3)*D[P[n - 1, x], x] + (3*x^3 + 12*x^4)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 32*n + 32)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 32*n + 32 + (24*n^2 - 102*n + 108)*x + (12*n^2 - 54*n + 60)*x^2)*v[i - 1] - ((4*n - 8)*x + (22*n - 44)*x^2 + (24*n - 66)*x^3)*deriv(v[i - 1], x) + (3*x^3 + 12*x^4)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 32*n + 32)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); 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 >= 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-a355ebab7822b2ca
• 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-a355ebab7822b2ca
• 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-a355ebab7822b2ca

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-a355ebab7822b2ca

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-8f881604bf5a9d75 root: theorem check

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

    • theorem_check three_term_recurrence e-3cffc405672d7710 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.

• A207607: Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.

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