A085881 → 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:           1;
  1:           2,            3;
  2:          12,           36,            27;
  3:         120,          540,           810,           405;
  4:        1680,        10080,         22680,         22680,           8505;
  5:       30240,       226800,        680400,       1020600,         765450,         229635;
  6:      665280,      5987520,      22453200,      44906400,       50519700,       30311820,        7577955;
  7:    17297280,    181621440,     817296480,    2043241200,     3064861800,     2758375620,     1379187810,      295540245;
  8:   518918400,   6227020800,   32691859200,   98075577600,   183891708000,   220670049600,   165502537200,    70929658800,   13299311025;
  9: 17643225600, 238183545600, 1429101273600, 5001854457600, 11254172529600, 16881258794400, 16881258794400, 10852237796400, 4069589173650, 678264862275;

The row polynomials satisfy P(n,x) = (4*n-2+(6*n-3)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n - 2 + (6*n - 3)*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] = (4*n - 2 + (6*n - 3)*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-0adda7d4e180ead0
• 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-0adda7d4e180ead0
• 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-0adda7d4e180ead0

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-0adda7d4e180ead0

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-1d211cea19207c02 root: theorem check

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

    • parent_provers root_domain e-088407700c450d53 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (23)

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

• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A085880 → basis A013610
• A196347 → basis A013610
• A156992 → basis A013610
• A356546 → basis A013610
• A178820 → basis A013610
• A120910 → basis A013609
• A038243 → basis A013611
• A038220 → D
• A038221 → basis A013610
• A038208 → basis A013610
• A038325 → basis A013610
• A038338 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A013620 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A028326 → basis A013610
• A120910 → x ↦ x/(x+1)
• A344565 → basis A013610
• A178821 → basis A013610
• A178822 → basis A013610
• A135838 → basis A013610

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