A116414 → reverse

OEIS matches ?

• A116415: 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 ?
• A116414: reversed triangle, 52 terms agree, 9 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:
  0: 1;
  1: 1,  4;
  2: 1,  8,  13;
  3: 1, 12,  42,   40;
  4: 1, 16,  87,  184,   121;
  5: 1, 20, 148,  496,   731,   364;
  6: 1, 24, 225, 1040,  2454,  2736,   1093;
  7: 1, 28, 318, 1880,  6170, 11064,   9844,   3280;
  8: 1, 32, 427, 3080, 13015, 32624,  46738,  34448,   9841;
  9: 1, 36, 552, 4704, 24381, 79044, 158724, 188208, 118101, 29524;

The row polynomials satisfy 2*n*P(n,x) = (2*n+(12*n-4)*x+(4*n-4)*x^2)*P(n-1,x) - (x+8*x^2+4*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 + (4*n - 4)*x^2)*P[n - 1, x] - (x + 8*x^2 + 4*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 + (4*n - 4)*x^2)*v[i - 1] - (x + 8*x^2 + 4*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:    13,      8,      1;
  3:    40,     42,     12,      1;
  4:   121,    184,     87,     16,     1;
  5:   364,    731,    496,    148,    20,     1;
  6:  1093,   2736,   2454,   1040,   225,    24,    1;
  7:  3280,   9844,  11064,   6170,  1880,   318,   28,   1;
  8:  9841,  34448,  46738,  32624, 13015,  3080,  427,  32,  1;
  9: 29524, 118101, 188208, 158724, 79044, 24381, 4704, 552, 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+(4*n-4)*x^2)*P(n-1,x) - (x+8*x^2+4*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 + (4*n - 4)*x^2)*P[n - 1, x] - (x + 8*x^2 + 4*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 + (4*n - 4)*x^2)*v[i - 1] - (x + 8*x^2 + 4*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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-18c37bd14c5a739f
• 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-18c37bd14c5a739f
• 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-18c37bd14c5a739f

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-18c37bd14c5a739f

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-894010de7c322dc8 root: theorem check

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

    • theorem_check three_term_recurrence e-7bedda0241d23786 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.

• A116414: Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

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