A344565 → 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:       3;
  1:      30,       15;
  2:     180,      225,        45;
  3:     840,     1995,       945,       105;
  4:    3360,    13650,     11550,      2940,       210;
  5:   12096,    79758,    107730,     47250,      7560,       378;
  6:   40320,   418950,    851130,    573300,    154350,     17010,      630;
  7:  126720,  2038410,   6008310,   5841990,   2356200,    429660,    34650,     990;
  8:  380160,  9362925,  39107475,  52827390,  29968785,   8045730,  1060290,   65340,   1485;
  9: 1098240, 41121795, 239478525, 438019725, 336035700, 124639515, 23873850, 2380950, 115830, 2145;

The row polynomials satisfy n*P(n,x) = (2*n+8+(n+4)*x)*P(n-1,x) + (n+4)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3.

P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (n + 4)*x)*P[n - 1, x] + (n + 4)*x*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (n + 4)*x)*v[i - 1] + (n + 4)*x*deriv(v[i - 1], x))/(n)); 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-19677b98e028e07a
• 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-19677b98e028e07a
• 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-19677b98e028e07a

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-19677b98e028e07a

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

  • premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-e2cde2e22920ce64 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • preserver_image reversal, shift 0 e-4dae17a76888d380

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-e2cde2e22920ce64 (expanded above)

Related sequences ?

Same fingerprint (19)

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

• A135838 → basis A008277
• A178822 → basis A008277
• A038221 → basis A008277
• A085881 → basis A008277
• A156992 → basis A008277
• A143494 → reverse
• A038325 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A038338 → basis A008277
• A085880 → basis A008277
• A178820 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A028326 → basis A008277
• A356546 → basis A008277
• A126351 → reverse
• A038208 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A196347 → basis A008277
• A178821 → 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