A038325 → 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:            22,             11;
  2:           484,            605,             121;
  3:         10648,          25289,           11979,            1331;
  4:        234256,         951665,          805255,          204974,           14641;
  5:       5153632,       33981761,        45899535,        20131375,         3221020,          161051;
  6:     113379904,     1178088065,      2393378911,      1612120510,       434032445,        47832147,        1771561;
  7:    2494357888,    40124085089,    118267640799,    114993796071,     46379466980,      8457432214,      682050985,      19487171;
  8:   54875873536,  1351532744705,   5645141131135,   7625602832694,   4325976577461,   1161396417258,   153052241034,    9431790764,    214358881;
  9: 1207269217792, 45204215184161, 263253069961695, 481504708240655, 369396085272060, 137013266480937, 26243957800830, 2617321937010, 127329175314, 2357947691;

The row polynomials satisfy P(n,x) = (22+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(22 + 11*x)*P[n - 1, x] + 11*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (22 + 11*x)*v[i - 1] + 11*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-f221360d3c3c712f
• 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-f221360d3c3c712f
• 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-f221360d3c3c712f

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-f221360d3c3c712f

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

  • premise (sequence): Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f7065dbe93a909bc root: theorem check

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

    • parent_provers periodic_product e-cbffb05414f7627e root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (19)

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

• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A126351 → reverse
• A356546 → basis A008277
• A038338 → basis A008277
• A085880 → basis A008277
• A178820 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A344565 → basis A008277
• A196347 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A038208 → basis A008277
• A028326 → basis A008277
• A156992 → basis A008277
• A038221 → basis A008277
• A085881 → basis A008277
• A135838 → basis A008277
• A143494 → reverse
• A178822 → 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