A085478 → x ↦ x+1

OEIS matches ?

• A001835: 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 ?
• A079935: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 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:    1;
  2:    2,    1;
  3:    5,    5,    1;
  4:   13,   19,    8,    1;
  5:   34,   65,   42,   11,    1;
  6:   89,  210,  183,   74,   14,    1;
  7:  233,  654,  717,  394,  115,   17,   1;
  8:  610, 1985, 2622, 1825,  725,  165,  20,  1;
  9: 1597, 5911, 9134, 7703, 3885, 1203, 224, 23, 1;

The row polynomials satisfy (2*n-3)*P(n,x) = (3*n-4+(n-1)*x)*P(n-1,x) + (5+6*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[((3*n - 4 + (n - 1)*x)*P[n - 1, x] + (5 + 6*x + x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = ((3*n - 4 + (n - 1)*x)*v[i - 1] + (5 + 6*x + 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 - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-b3424874909a70f6
• 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-b3424874909a70f6, e-c1b3c12b9e216f54
• 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-b3424874909a70f6, e-c1b3c12b9e216f54
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-6f13642fbb28ea1d, e-9f91faf8953a1d7a, e-b3424874909a70f6, e-c1b3c12b9e216f54
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-6f13642fbb28ea1d
• 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-6f13642fbb28ea1d, e-9f91faf8953a1d7a
• 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-6f13642fbb28ea1d, e-9f91faf8953a1d7a
• 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-c1b3c12b9e216f54
• 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-9f91faf8953a1d7a

Proof chains

5 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-9f91faf8953a1d7a

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-cec419fd822ca749 root: theorem check

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

    • theorem_check three_term_recurrence e-59d957f5dca11392 root: theorem check

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

• basis_map A007318 e-c1b3c12b9e216f54

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-2580795600db03bb root: theorem check

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

    • theorem_check three_term_recurrence e-365c9804d88a701d root: theorem check

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

• definition_equality e-a08c2f082979b173 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-6f13642fbb28ea1d

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)
    • ↵ theorem_check three_term_recurrence e-59d957f5dca11392 (expanded above)
• preserver_image affine, shift 0 e-b3424874909a70f6

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0)
    • ↵ theorem_check three_term_recurrence e-365c9804d88a701d (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A054142 → x ↦ x/(x+1)

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