narayana → γ

OEIS matches ?

• A001006: 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 ?
• A168049: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A086246: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A055151: triangle, 61 terms agree, 14 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 992 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: 1;
   3: 1,  1;
   4: 1,  3;
   5: 1,  6,   2;
   6: 1, 10,  10;
   7: 1, 15,  30,   5;
   8: 1, 21,  70,  35;
   9: 1, 28, 140, 140,  14;
  10: 1, 36, 252, 420, 126;

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

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1)*P[n - 1, x] - (n - 2 - (4*n - 8)*x)*P[n - 2, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n - 1)*v[i - 1] - (n - 2 - (4*n - 8)*x)*v[i - 2])/(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:
  0: 1;
  1: 1;
  2: 1,  1;
  3: 1,  3;
  4: 1,  6,   2;
  5: 1, 10,  10;
  6: 1, 15,  30,   5;
  7: 1, 21,  70,  35;
  8: 1, 28, 140, 140,  14;
  9: 1, 36, 252, 420, 126;

The row polynomials satisfy (n+2)*P(n,x) = (2*n+1)*P(n-1,x) - (n-1-(4*n-4)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*P[n - 1, x] - (n - 1 - (4*n - 4)*x)*P[n - 2, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*v[i - 1] - (n - 1 - (4*n - 4)*x)*v[i - 2])/(n + 2)); 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 >= 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-dbb6454df235fcfe
• 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-dbb6454df235fcfe
• 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-dbb6454df235fcfe

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

• gamma_polynomial scale 1, shift 0 e-dbb6454df235fcfe

  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

  • 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 (4)

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

• A011973 → ⊙ C(n,k) → D
• A059344 → ⊙ 1/k! (shift -1) → D
• A055151: Triangular array of Motzkin polynomial coefficients.
• A132812 → γ

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