A038259 → basis A013611

OEIS matches ?

• A009975: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:         11,          20;
  2:        121,         440,          400;
  3:       1331,        7260,        13200,          8000;
  4:      14641,      106480,       290400,        352000,        160000;
  5:     161051,     1464100,      5324000,       9680000,       8800000,       3200000;
  6:    1771561,    19326120,     87846000,     212960000,     290400000,     211200000,      64000000;
  7:   19487171,   248018540,   1352828400,    4099480000,    7453600000,    8131200000,    4928000000,    1280000000;
  8:  214358881,  3117947360,  19841483200,   72150848000,  163979200000,  238515200000,  216832000000,  112640000000,   25600000000;
  9: 2357947691, 38584598580, 280615262400, 1190488992000, 3246788160000, 5903251200000, 7155456000000, 5575680000000, 2534400000000, 512000000000;

The row polynomials satisfy P(n,x) = (11+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(11 + 20*x)*P[n - 1, 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] = (11 + 20*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-17e7dceca4b24148
• 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-17e7dceca4b24148
• 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-17e7dceca4b24148

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-17e7dceca4b24148

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-0a7984b7b9125974 root: theorem check

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

    • parent_provers periodic_product e-7949976749bfedcf root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• A038301 → basis A007318
• A038336 → basis A013611

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