narayana → basis A013609

OEIS matches ?

• A007564: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:     2,      2;
   3:     5,     10,      4;
   4:    14,     42,     36,       8;
   5:    42,    168,    224,     112,      16;
   6:   132,    660,   1200,     960,     320,      32;
   7:   429,   2574,   5940,    6600,    3600,     864,     64;
   8:  1430,  10010,  28028,   40040,   30800,   12320,   2240,    128;
   9:  4862,  38896, 128128,  224224,  224224,  128128,  39424,   5632,   256;
  10: 16796, 151164, 572832, 1188096, 1467648, 1100736, 489216, 119808, 13824, 512;

The row polynomials satisfy (n+1)*P(n,x) = -(4*n-8)*x^2*P(n-2,x) + (4*n-2+(4*n-2)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8)*x^2*P[n - 2, x] + (4*n - 2 + (4*n - 2)*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] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(4*n - 8)*x^2*v[i - 2] + (4*n - 2 + (4*n - 2)*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-2f325d6b3b1beb16
• 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-2f325d6b3b1beb16
• 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-2f325d6b3b1beb16
• 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-2f325d6b3b1beb16

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 A013609 e-2f325d6b3b1beb16

  Geometric basis map (elementary) polytool ?

  • premise (sequence): narayana
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check

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

    • preserver_image reversal, shift 0 e-29ef477df4172e22

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): narayana
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A001263 → basis A013609

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