A038337 → basis A013610

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,              1518,               1089;
  1:           12167,             52371,              75141,              35937;
  2:          279841,           1606044,            3456486,            3306204,             1185921;
  3:         6436343,          46173765,          132498630,          190106730,           136380915,            39135393;
  4:       148035889,        1274395914,         4571202735,         8744909580,          9410283135,          5400684234,          1291467969;
  5:      3404825447,       34196290359,       147192728067,       351982610595,        505018528245,        434755080837,        207926343009,         42618442977;
  6:     78310985281,      898873918008,      4513910327388,     12952960069896,      23230852299270,      26664978291336,      19129223556828,       7841793507768,       1406408618241;
  7:   1801152661463,    23258362628457,    133482776824188,    446877122411412,     961757285189778,    1379912626576638,    1319916425421132,     811625628053988,     291126583975887,      46411484401953;
  8:  41426511213649,   594380378282790,   3837629833695405,  14683105450660680,   36867362598941490,   63475980822525348,   75895194461715090,   62224631484139080,   33479557157227005,   10674641412449190,   1531578985264449;
  9: 952809757913927, 15037823570554587, 107880038658326385, 464353209877144005, 1332491819647456710, 2676570524683152174, 3840296839762783554, 3935707941371796810, 2823442653592810755, 1350342138674822535, 387489483271905597, 50542106513726817;

The row polynomials satisfy P(n,x) = (23+33*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 1518*x + 1089*x^2.

P[0, x_] := 529 + 1518*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 33*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 + 1518*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 33*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-50822203bd5493c4
• 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-50822203bd5493c4
• 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-50822203bd5493c4

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 A013610 e-50822203bd5493c4

  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