A008288 → 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:    2,     1;
   3:    5,     6,      1;
   4:   12,    27,     11,      1;
   5:   29,   110,     80,     17,      1;
   6:   70,   425,    490,    180,     24,      1;
   7:  169,  1594,   2735,   1538,    346,     32,     1;
   8:  408,  5871,  14429,  11635,   3914,    600,    41,    1;
   9:  985, 21378,  73386,  81491,  38387,   8652,   967,   51,  1;
  10: 2378, 77269, 364080, 541810, 343018, 106465, 17275, 1475, 62, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + x*d/dx P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx 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[(1 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
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] = (1 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (2 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-48d65c291ee1d51e
• 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-48d65c291ee1d51e
• 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-48d65c291ee1d51e

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-48d65c291ee1d51e

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

  • premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-b176436450506e78 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

    • preserver_image reversal, shift 0 e-73f1cbc14f4fad08

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang_derivative_lag e-b176436450506e78 (expanded above)
        • basis_map A008288 e-796ce5140a0b4b04

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-b8e0a0fa3ba7bf57

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
    • ↵ basis_map A008288 e-796ce5140a0b4b04 (expanded above)
    • ↵ basis_map A008288 e-b8e0a0fa3ba7bf57 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• delannoy → basis A008277

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
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