A064861 → 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:     6,       9,        2;
  1:    18,      57,       32,         4;
  2:    36,     189,      185,        52,         4;
  3:   108,    1017,     1673,       838,       148,         8;
  4:   216,    3159,     7709,      5863,      1726,       204,        8;
  5:   648,   16227,    60281,     68185,     30712,      6104,      528,       16;
  6:  1296,   49329,   257351,    401206,    252457,     73440,    10328,      672,      16;
  7:  3888,  249237,  1900115,   4125556,   3579439,   1459634,   301800,    32080,    1648,     32;
  8:  7776,  751599,  7849697,  22527895,  25602190,  13796877,  3874034,   590520,   48560,   2000,   32;
  9: 23328, 3773547, 56451077, 218450449, 326679880, 230563781, 85704264, 17786908, 2103680, 139120, 4736, 64;

The row polynomials satisfy P(n,x) = (6+9*x+2*x^2)*P(n-2,x) + (9*x+4*x^2)*d/dx P(n-2,x) + 2*x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 6 + 9*x + 2*x^2, P(1,x) = 18 + 57*x + 32*x^2 + 4*x^3.

P[0, x_] := 6 + 9*x + 2*x^2;
P[1, x_] := 18 + 57*x + 32*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[(6 + 9*x + 2*x^2)*P[n - 2, x] + (9*x + 4*x^2)*D[P[n - 2, x], x] + 2*x^2*D[P[n - 2, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 9*x + 2*x^2; v[2] = 18 + 57*x + 32*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 9*x + 2*x^2)*v[i - 2] + (9*x + 4*x^2)*deriv(v[i - 2], x) + 2*x^2*deriv(deriv(v[i - 2], x), 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-a9b5217f78edb4be
• 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-a9b5217f78edb4be
• 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-a9b5217f78edb4be

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-a9b5217f78edb4be

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

  • premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-bbffc3f85cdc52e2 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