A038226 → 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:          121,             240,               64;
  1:         1331,            5528,             3648,               512;
  2:        14641,          115680,           142720,             47104,              4096;
  3:       161051,         2315048,          4778880,           2790400,            552960,             32768;
  4:      1771561,        45274320,        147550144,         135895040,          46100480,           6094848,           262144;
  5:     19487171,       874384568,       4346048448,        5936727552,        3069480960,         679641088,         64225280,          2097152;
  6:    214358881,     16769204160,     124338384640,      242553851904,      179481501696,       59217543168,       9226420224,        654311424,        16777216;
  7:   2357947691,    320329750088,    3491290018560,     9484091893760,     9658135388160,     4455946715136,    1018099138560,     117650227200,      6492782592,      134217728;
  8:  25937424601,   6105128833200,   96827468501824,   359873536430080,   491172556840960,   304518365577216,   95715422896128,   16027358330880,   1428160512000,    63082332160,   1073741824;
  9: 285311670611, 116204947227608, 2663182680214848, 13370193523067392, 23999408235601920, 19459817099165696, 8083356875489280, 1839556391337984, 235330905047040, 16661117665280, 602369163264, 8589934592;

The row polynomials satisfy P(n,x) = (11+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 121 + 240*x + 64*x^2.

P[0, x_] := 121 + 240*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 8*x)*P[n - 1, x] + 8*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] = 121 + 240*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 8*x)*v[i - 1] + 8*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-fc0676c2630bb222
• 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-fc0676c2630bb222
• 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-fc0676c2630bb222

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-fc0676c2630bb222

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

  • premise (sequence): Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-fdf6b848b46b3343 root: theorem check

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

    • parent_provers periodic_product e-711ca5eff5ba56ea root: theorem check

      Parent-row provers of polytool polytool ?

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