A082985 → reverse

OEIS matches ?

• A093960: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A111125: 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 = 0 ?
• A135064: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A082985: 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 ?
• A002878: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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:  3,   1;
  2:  5,   5,    1;
  3:  7,  14,    7,    1;
  4:  9,  30,   27,    9,    1;
  5: 11,  55,   77,   44,   11,    1;
  6: 13,  91,  182,  156,   65,   13,   1;
  7: 15, 140,  378,  450,  275,   90,  15,   1;
  8: 17, 204,  714, 1122,  935,  442, 119,  17,  1;
  9: 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1;

The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+n*x)*P(n-1,x) + (4*x+x^2)*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 + 1 + n*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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 + 1 + n*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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: 1,  3;
  2: 1,  5,   5;
  3: 1,  7,  14,   7;
  4: 1,  9,  27,  30,    9;
  5: 1, 11,  44,  77,   55,   11;
  6: 1, 13,  65, 156,  182,   91,   13;
  7: 1, 15,  90, 275,  450,  378,  140,   15;
  8: 1, 17, 119, 442,  935, 1122,  714,  204,  17;
  9: 1, 19, 152, 665, 1729, 2717, 2508, 1254, 285, 19;

Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-1)*P(n,x) = (2*n+1+n*x)*P(n-1,x) + (4*x+x^2)*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 + 1 + n*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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 + 1 + n*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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)

Triangle begins:
  0:  1;
  1:  3,   1;
  2:  5,   5,    1;
  3:  7,  14,    7,    1;
  4:  9,  30,   27,    9,    1;
  5: 11,  55,   77,   44,   11,    1;
  6: 13,  91,  182,  156,   65,   13,   1;
  7: 15, 140,  378,  450,  275,   90,  15,   1;
  8: 17, 204,  714, 1122,  935,  442, 119,  17,  1;
  9: 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1;

The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+n*x)*P(n-1,x) + (4*x+x^2)*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 + 1 + n*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[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 + 1 + n*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-41a825a32145fbe0
• 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-41a825a32145fbe0
• 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-41a825a32145fbe0

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-41a825a32145fbe0

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4fa62233b528a616 root: theorem check

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

Related sequences ?

Same fingerprint (4)

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

• A111125 → reverse
• A082985: Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
• A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
• A091042 → basis A013611

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