A122431 → 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: 8,    19,      9,       1;
  1: 8,    46,     46,      13,        1;
  2: 8,   100,    184,      98,       18,        1;
  3: 8,   208,    652,     576,      188,       24,       1;
  4: 8,   424,   2164,    2956,     1516,      332,      31,       1;
  5: 8,   856,   6916,   13988,    10536,     3508,     549,      39,      1;
  6: 8,  1720,  21604,   62868,    66668,    31584,    7351,     861,     48,     1;
  7: 8,  3448,  66532,  273076,   396208,   256172,   83041,   14239,   1293,    58,    1;
  8: 8,  6904, 203044, 1158836,  2254116,  1933240,  837459,  196953,  25876,  1873,   69,  1;
  9: 8, 13816, 616036, 4838388, 12429416, 13853556, 7795453, 2413083, 429837, 44606, 2632, 81, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8 + 19*x + 9*x^2 + x^3.

P[0, x_] := 8 + 19*x + 9*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 8 + 19*x + 9*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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-22aa1241157a3ec8
• 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-22aa1241157a3ec8
• 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-22aa1241157a3ec8

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-22aa1241157a3ec8

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

  • premise (sequence): Riordan array ((1+x)^3,x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-73f8c6c9e8d957af root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [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