A038337 → basis A013611

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,              2024,               1936;
  1:           12167,             69828,             133584,               85184;
  2:          279841,           2141392,            6144864,             7836928,             3748096;
  3:         6436343,          61565020,          235553120,           450623360,           431031040,            164916224;
  4:       148035889,        1699194552,         8126582640,         20728674560,         29741141760,          22758438912,           7256313856;
  5:      3404825447,       45595053812,       261675961008,        834329151040,       1596107941120,        1832054332416,        1168266530816,         319277809664;
  6:     78310985281,     1198498557344,      8024729470912,      30703312758272,      73420965291520,      112365999054848,      107480520835072,       58747116978176,       14048223625216;
  7:   1801152661463,    31011150171276,    237302714354112,    1059264290160384,    3039627963068928,     5814940451088384,     7416155937619968,     6080326607241216,     2907982290419712,      618121839509504;
  8:  41426511213649,   792507171043720,   6822453037680720,   34804398105269760,  116519071917642240,   267487260750065664,   426428966413148160,   466158373221826560,   334417963398266880,   142168023087185920,   27197360938418176;
  9: 952809757913927, 20050431427406116, 191786735392580240, 1100689090079156160, 4211332170737640960, 11279046161627768832, 21577305700505296896, 29484517106280529920, 28202581579920506880, 17984254920529018880, 6880932317419798528, 1196683881290399744;

The row polynomials satisfy P(n,x) = (23+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 2024*x + 1936*x^2.

P[0, x_] := 529 + 2024*x + 1936*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 44*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 + 2024*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 44*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-03b8863ee0776575
• 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-03b8863ee0776575
• 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-03b8863ee0776575

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 A013611 e-03b8863ee0776575

  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