A038335 → 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:            1;
  1:           21,             36;
  2:          441,           1512,           1296;
  3:         9261,          47628,          81648,           46656;
  4:       194481,        1333584,        3429216,         3919104,         1679616;
  5:      4084101,       35006580,      120022560,       205752960,       176359680,         60466176;
  6:     85766121,      882165816,     3780710640,      8641624320,     11110659840,       7618738176,       2176782336;
  7:   1801088541,    21613062492,   111152892816,    317579693760,    544422332160,     559977255936,     319987003392,      78364164096;
  8:  37822859361,   518713499808,  3112280998848,  10670677710336,  22865737950720,   31358726332416,   26878908284928,   13165179568128,   2821109907456;
  9: 794280046581, 12254606432964, 84031586968896, 336126347875584, 864324894537216, 1481699819206656, 1693371221950464, 1244109469188096, 533189772509184, 101559956668416;

The row polynomials satisfy P(n,x) = (21+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(21 + 36*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] = (21 + 36*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-3428d0ea6005b0ef
• 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-3428d0ea6005b0ef
• 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-3428d0ea6005b0ef

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-3428d0ea6005b0ef

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-03786f96284c3301 root: theorem check

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

    • parent_provers periodic_product e-2dcf274e8e5380c5 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

• A038284 → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• A013613 → basis A013609

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