A038337 → basis A007318

OEIS matches ?

• A009978: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,              506,               121;
  1:           12167,            17457,              8349,              1331;
  2:          279841,           535348,            384054,            122452,             14641;
  3:         6436343,         15391255,          14722070,           7040990,           1683715,            161051;
  4:       148035889,        424798638,         507911415,         323885540,         116176335,          22225038,          1771561;
  5:      3404825447,      11398763453,       16354747563,       13036392985,        6234796645,        1789115559,        285221321,         19487171;
  6:     78310985281,     299624639336,      501545591932,      479739261848,      286800645670,      109732420952,      26240361532,       3585639464,       214358881;
  7:   1801152661463,    7752787542819,    14831419647132,    16551004533756,    11873546730738,     5678652784266,    1810584945708,     371113684524,     44372288367,     2357947691;
  8:  41426511213649,  198126792760930,   426403314855045,   543818720394840,   455152624678290,   261218028076236,  104108634378210,   28452049146840,   5102813162205,   542327968930,   25937424601;
  9: 952809757913927, 5012607856851529, 11986670962036265, 17198267032486815, 16450516291943910, 11014693517214618, 5267896899537426, 1799592108537630, 430337243345955, 68604488069645, 6562168424053, 285311670611;

The row polynomials satisfy P(n,x) = (23+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 506*x + 121*x^2.

P[0, x_] := 529 + 506*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 11*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 529 + 506*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 11*x)*v[i - 1]); 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-347b6efdbfa5233f
• 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-347b6efdbfa5233f
• 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-347b6efdbfa5233f

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 A007318 e-347b6efdbfa5233f

  Geometric basis map (elementary) polytool ?

  • 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