A154227 → 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:
  0:       1;
  1:       2,         1;
  2:      10,        11,         1;
  3:      40,        83,        25,         1;
  4:     230,       769,       393,        45,        1;
  5:    1300,      6863,      5659,      1227,       72,        1;
  6:    9040,     71393,     84043,     27838,     3059,      107,      1;
  7:   64880,    764155,   1265805,    603645,   103324,     6616,    151,     1;
  8:  536560,   9189515,  20385865,  13076805,  3164574,   316181,  12954,   205,   1;
  9: 4641520, 115730555, 341619325, 288191065, 93678824, 13243941, 841828, 23544, 270, 1;

The row polynomials satisfy P(n,x) = ((1/2)*n^2+(3/2)*n+1+((1/2)*n^2+(3/2)*n+1)*x)*P(n-2,x) + ((1/2)*n^2+(3/2)*n+1)*x*d/dx P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[((1/2)*n^2 + (3/2)*n + 1 + ((1/2)*n^2 + (3/2)*n + 1)*x)*P[n - 2, x] + ((1/2)*n^2 + (3/2)*n + 1)*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, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((1/2)*n^2 + (3/2)*n + 1 + ((1/2)*n^2 + (3/2)*n + 1)*x)*v[i - 2] + ((1/2)*n^2 + (3/2)*n + 1)*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 - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, 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-d61a33ddedddac02
• Coefficients of one sign. Every nonzero row, n >= 0: 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-d61a33ddedddac02
• Hadamard kernel (Malo). Rows P_n, n >= 0, 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-d61a33ddedddac02

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-d61a33ddedddac02

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

  • premise (sequence): Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-f727ccd9a8e875c3 root: theorem check

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

Related sequences ?

No candidate equalities or related sequences.

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