A063007 → basis A013610

OEIS matches ?

• A084769: 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:
   1:       1;
   2:       3,        6;
   3:      13,       54,        54;
   4:      63,      396,       810,        540;
   5:     321,     2700,      8370,      11340,       5670;
   6:    1683,    17730,     73710,     151200,     153090,      61236;
   7:    8989,   113778,    593460,    1632960,    2500470,    2020788,     673596;
   8:   48639,   718872,   4511052,   15581160,   31995810,   39068568,   26270244,    7505784;
   9:  265729,  4491288,  32942700,  136964520,  353076570,  577945368,  586702116,  337760280,   84440070;
  10: 1462563, 27823230, 233566740, 1135632960, 3524591070, 7241830596, 9851341500, 8556593760, 4306443570, 956987460;

The row polynomials satisfy -(n-1)*P(n,x) = (n-2)*P(n-2,x) - (6*n-9+(12*n-18)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 6*x.

P[1, x_] := 1;
P[2, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*P[n - 2, x] - (6*n - 9 + (12*n - 18)*x)*P[n - 1, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 6*x; for(n = 3, N, my(i = n); v[i] = ((n - 2)*v[i - 2] - (6*n - 9 + (12*n - 18)*x)*v[i - 1])/(-n + 1)); 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)

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-c0d1bf6258b5bb5a
• 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-c0d1bf6258b5bb5a
• 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-c0d1bf6258b5bb5a
• 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-c0d1bf6258b5bb5a

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

• basis_map A013610 e-c0d1bf6258b5bb5a

  Geometric basis map (elementary) polytool ?

  • premise (sequence): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-18de01688f1c88b0 root: theorem check

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

    • theorem_check three_term_recurrence e-93f30859d3148f4f root: theorem check

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

    • preserver_image reversal, shift 0 e-00d47151fcd2720e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-74c20febd3423e2e root: theorem check

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

        • preserver_image reversal, shift 0 e-5047c71cb920acb6

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-18de01688f1c88b0 (expanded above)
            • ↵ theorem_check three_term_recurrence e-93f30859d3148f4f (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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