A091042 → reverse

OEIS matches ?

• A155543: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A347478: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A103327: 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 ?
• A000302: 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 ?
• A091042: 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 = 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:
   1:  1;
   2:  3,   1;
   3:  5,  10,     1;
   4:  7,  35,    21,     1;
   5:  9,  84,   126,    36,     1;
   6: 11, 165,   462,   330,    55,     1;
   7: 13, 286,  1287,  1716,   715,    78,     1;
   8: 15, 455,  3003,  6435,  5005,  1365,   105,    1;
   9: 17, 680,  6188, 19448, 24310, 12376,  2380,  136,   1;
  10: 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1;

The row polynomials satisfy (2*n-3)*P(n,x) = (2*n-1+(6*n-11)*x)*P(n-1,x) + (4*x-4*x^2)*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[((2*n - 1 + (6*n - 11)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = ((2*n - 1 + (6*n - 11)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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)

Triangle begins:
  0: 1;
  1: 1,   3;
  2: 1,  10,    5;
  3: 1,  21,   35,     7;
  4: 1,  36,  126,    84,     9;
  5: 1,  55,  330,   462,   165,    11;
  6: 1,  78,  715,  1716,  1287,   286,    13;
  7: 1, 105, 1365,  5005,  6435,  3003,   455,    15;
  8: 1, 136, 2380, 12376, 24310, 19448,  6188,   680,  17;
  9: 1, 171, 3876, 27132, 75582, 92378, 50388, 11628, 969, 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+(6*n-5)*x)*P(n-1,x) + (4*x-4*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 + (6*n - 5)*x)*P[n - 1, x] + (4*x - 4*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 + (6*n - 5)*x)*v[i - 1] + (4*x - 4*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,  10,     1;
  3:  7,  35,    21,     1;
  4:  9,  84,   126,    36,     1;
  5: 11, 165,   462,   330,    55,     1;
  6: 13, 286,  1287,  1716,   715,    78,     1;
  7: 15, 455,  3003,  6435,  5005,  1365,   105,    1;
  8: 17, 680,  6188, 19448, 24310, 12376,  2380,  136,   1;
  9: 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1;

The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(6*n-5)*x)*P(n-1,x) + (4*x-4*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 + (6*n - 5)*x)*P[n - 1, x] + (4*x - 4*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 + (6*n - 5)*x)*v[i - 1] + (4*x - 4*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 >= 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-e14f5dbc35a40145
• 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-e14f5dbc35a40145
• 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-e14f5dbc35a40145

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-e14f5dbc35a40145

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-1c0df90858517049 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.

• A103327 → reverse
• A103327: Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
• A091042: Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
• A086645 → D

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