A104684 → basis A008277

OEIS matches ?

No OEIS matches.

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,        1;
   3:      13,        9,        1;
   4:      63,       73,       18,        1;
   5:     321,      565,      235,       30,       1;
   6:    1683,     4261,     2660,      575,      45,       1;
   7:    8989,    31647,    27811,     8960,    1190,      63,      1;
   8:   48639,   232849,   276794,   123991,   24430,    2198,     84,    1;
   9:  265729,  1703205,  2666203,  1590414,  432201,   57498,   3738,  108,   1;
  10: 1462563, 12412981, 25105500, 19368175, 6937917, 1265817, 121380, 5970, 135, 1;

The row polynomials satisfy -(n-1)*P(n,x) = (n-2+(3*n-6)*x+(n-2)*x^2)*P(n-2,x) + ((3*n-6)*x+(2*n-4)*x^2)*d/dx P(n-2,x) + (n-2)*x^2*d^2/dx^2 P(n-2,x) - (6*n-9+(2*n-3)*x)*P(n-1,x) - (2*n-3)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.

P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (3*n - 6)*x + (n - 2)*x^2)*P[n - 2, x] + ((3*n - 6)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] + (n - 2)*x^2*D[P[n - 2, x], {x, 2}] - (6*n - 9 + (2*n - 3)*x)*P[n - 1, x] - (2*n - 3)*x*D[P[n - 1, x], 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 + x; for(n = 3, N, my(i = n); v[i] = ((n - 2 + (3*n - 6)*x + (n - 2)*x^2)*v[i - 2] + ((3*n - 6)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) + (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) - (6*n - 9 + (2*n - 3)*x)*v[i - 1] - (2*n - 3)*x*deriv(v[i - 1], x))/(-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-907fec0ae54b396c
• 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-907fec0ae54b396c
• 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-907fec0ae54b396c

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 A008277 e-907fec0ae54b396c

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • 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 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 (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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023