narayana → basis A007318

OEIS matches ?

• A001003: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A126216: 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 = 1 ?
• A033282: reversed 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 = 3 ?

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:     2,     1;
   3:     5,     5,      1;
   4:    14,    21,      9,      1;
   5:    42,    84,     56,     14,     1;
   6:   132,   330,    300,    120,    20,     1;
   7:   429,  1287,   1485,    825,   225,    27,    1;
   8:  1430,  5005,   7007,   5005,  1925,   385,   35,   1;
   9:  4862, 19448,  32032,  28028, 14014,  4004,  616,  44,  1;
  10: 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1;

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

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

Triangle begins:
   3: 1;
   4: 1,  2;
   5: 1,  5,   5;
   6: 1,  9,  21,   14;
   7: 1, 14,  56,   84,    42;
   8: 1, 20, 120,  300,   330,   132;
   9: 1, 27, 225,  825,  1485,  1287,    429;
  10: 1, 35, 385, 1925,  5005,  7007,   5005,   1430;
  11: 1, 44, 616, 4004, 14014, 28028,  32032,  19448,  4862;
  12: 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796;

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = -(n-4)*x^2*P(n-2,x) + (4*n-10+(2*n-5)*x)*P(n-1,x) for n >= 5, with P(3,x) = 1, P(4,x) = 2 + x.

P[3, x_] := 1;
P[4, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*x^2*P[n - 2, x] + (4*n - 10 + (2*n - 5)*x)*P[n - 1, x])/(n - 1)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 2]], {n, 3, 12}] // Flatten

P(N) = {my(v = vector(N - 2)); v[1] = 1; v[2] = 2 + x; for(n = 5, N, my(i = n - 2); v[i] = (-(n - 4)*x^2*v[i - 2] + (4*n - 10 + (2*n - 5)*x)*v[i - 1])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};
T(12)

Triangle begins:
   1:     1;
   2:     2,     1;
   3:     5,     5,      1;
   4:    14,    21,      9,      1;
   5:    42,    84,     56,     14,     1;
   6:   132,   330,    300,    120,    20,     1;
   7:   429,  1287,   1485,    825,   225,    27,    1;
   8:  1430,  5005,   7007,   5005,  1925,   385,   35,   1;
   9:  4862, 19448,  32032,  28028, 14014,  4004,  616,  44,  1;
  10: 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1;

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

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2)*x^2*P[n - 2, x] + (4*n - 2 + (2*n - 1)*x)*P[n - 1, x])/(n + 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(n - 2)*x^2*v[i - 2] + (4*n - 2 + (2*n - 1)*x)*v[i - 1])/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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-58b8cdf8af6ac600
• 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-58b8cdf8af6ac600, e-9c1368c49a29af4d
• 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-58b8cdf8af6ac600, e-9c1368c49a29af4d
• 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-58b8cdf8af6ac600, e-9c1368c49a29af4d
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-9c1368c49a29af4d

Proof chains

3 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-58b8cdf8af6ac600

  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)
• definition_equality e-d592448ae450a1b6 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-9c1368c49a29af4d

  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)
    • ↵ preserver_image reversal, shift 0 e-29ef477df4172e22 (expanded above)

Related sequences ?

Same fingerprint (6)

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

• A063007 → D
• A132812 → x ↦ x/(x+1)
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but …
• A132812 → x ↦ x+1
• narayana → x ↦ x/(x+1)
• A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

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