A038305 → basis A013609

OEIS matches ?

• A001029: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:           169,           156,             36;
  1:          2197,          3042,           1404,            216;
  2:         28561,         52728,          36504,          11232,           1296;
  3:        371293,        856830,         790920,         365040,          84240,           7776;
  4:       4826809,      13366548,       15422940,        9491040,        3285360,         606528,         46656;
  5:      62748517,     202725978,      280697508,      215921160,       99655920,       27597024,       4245696,        279936;
  6:     815730721,    3011928816,     4865423472,     4491160128,     2591053920,      956696832,     220776192,      29113344,      1679616;
  7:   10604499373,   44049458934,    81322078032,    87577622496,    60630661728,    27983382336,    8610271488,    1703130624,    196515072,    10077696;
  8:  137858491849,  636269962380,  1321483768020,  1626441560640,  1313664337440,   727567940736,  279833823360,   73802327040,  12773479680,  1310100480,   60466176;
  9: 1792160394037, 9098660462034, 20996908758540, 29072642896440, 26836285750560, 17340369254208, 8003247348096, 2638433191680, 608869198080, 93672184320, 8646663168, 362797056;

The row polynomials satisfy P(n,x) = (13+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 156*x + 36*x^2.

P[0, x_] := 169 + 156*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(13 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 169 + 156*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (13 + 6*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-54bb49968ded72d0
• 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-54bb49968ded72d0
• 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-54bb49968ded72d0

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 A013609 e-54bb49968ded72d0

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-e186153c671b61f0 root: theorem check

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

    • parent_provers periodic_product e-f733aee5869de636 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