A272866 → 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:       5,          6,            1;
  2:      25,         96,           72,            16,              1;
  3:     125,       1206,         2161,          1280,            305,             30,              1;
  4:     625,      14016,        50832,         59936,          30026,           7200,            856,             48,              1;
  5:    3125,     157926,      1078561,       2260480,        2001130,         876300,         206002,          26880,           1925,            70,            1;
  6:   15625,    1755936,     21759192,      76497456,      110250651,       78472800,       30648072,        6976656,         953427,         78400,         3760,          96,          1;
  7:   78125,   19409046,    427487761,    2437774080,     5460005355,     5936010570,     3541828563,     1251060480,      274104735,      38222450,      3417827,      193536,       6657,       126,        1;
  8:  390625,  213968256,   8277617952,   74989734976,   253540996276,   405358430400,   351906002448,   181033307424,    58440454278,   12288505600,   1721206720,   162115968,   10228148,    423360,    10960,    160,     1;
  9: 1953125, 2355994566, 158986877761, 2257692462080, 11303335284980, 25919277354840, 31724602056484, 22851630835200, 10342581667230, 3076517535140, 620151630278, 86382870528, 8398136756, 570143448, 26745828, 844800, 17061, 198, 1;

The row polynomials satisfy P(n,x) = (5+6*x+x^2)*P(n-1,x) + (6*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 6*x + x^2)*P[n - 1, x] + (6*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + 6*x + x^2)*v[i - 1] + (6*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], 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-8606f224a066b2e0
• 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-8606f224a066b2e0
• 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-8606f224a066b2e0

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-8606f224a066b2e0

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

  • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-0e058c428dfdfeb6

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)

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