A038337 → 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:             529,               627,                121;
  1:           12167,             27137,              12342,                1331;
  2:          279841,           1056495,             853897,              210298,               14641;
  3:         6436343,          38999081,           50046810,            21169555,             3294225,              161051;
  4:       148035889,        1396768527,         2681096341,          1736009990,           453578180,            48798453,            1771561;
  5:      3404825447,       49118524697,       136013789142,        126708619191,         49485847950,          8795639314,          694451912,           19487171;
  6:     78310985281,     1707482919615,      6660924283057,       8591834355258,       4709346623751,       1230404193942,       158558252622,         9587688132,         214358881;
  7:   1801152661463,    58915840105001,    318523904853330,     554412891008075,     410036401699155,     147774339988737,     27646130616720,      2702909592042,      129258405243,      2357947691;
  8:  41426511213649,  2022951242846127,  14981649959554861,   34550884849838830,   33570980714932210,   16036798937812191,   4086023364764187,    574398395988120,    44079688494435,   1709512075975,    25937424601;
  9: 952809757913927, 69236033880118457, 696426711851276142, 2099647701146077951, 2629315441248685200, 1620151105013605208, 540060867779946744, 102386096611218057, 11211227778751605, 693437046707735, 22254310307658, 285311670611;

The row polynomials satisfy P(n,x) = (23+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 529 + 627*x + 121*x^2.

P[0, x_] := 529 + 627*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 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] = 529 + 627*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 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-29e8dbf6785f009b
• 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-29e8dbf6785f009b
• 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-29e8dbf6785f009b

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-29e8dbf6785f009b

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-37ab130d55b994ea root: theorem check

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

    • basis_map A007318 e-107a218257e6f127

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle of coefficients in expansion of (1+11x)^n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-5d158048399db1f0 root: theorem check

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

        • parent_provers periodic_product e-70681fa17f5604ff root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-9aedabfb67a4638b 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