A038284 → basis A007318

OEIS matches ?

• A009964: 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:           196,            168,             36;
  1:          2744,           3528,           1512,            216;
  2:         38416,          65856,          42336,          12096,           1296;
  3:        537824,        1152480,         987840,         423360,          90720,           7776;
  4:       7529536,       19361664,       20744640,       11854080,        3810240,         653184,          46656;
  5:     105413504,      316240512,      406594944,      290424960,      124467840,       32006016,        4572288,        279936;
  6:    1475789056,     5059848192,     7589772288,     6505519104,     3485099520,     1194891264,      256048128,      31352832,      1679616;
  7:   20661046784,    79692609024,   136615901184,   136615901184,    87824507904,    37639074816,    10754021376,    1975228416,    211631616,     10077696;
  8:  289254654976,  1239662807040,  2390778270720,  2732318023680,  2049238517760,  1053894094848,   376390748160,   92177326080,  14814213120,   1410877440,   60466176;
  9: 4049565169664, 19090807228416, 40908872632320, 52597121955840, 45083247390720, 27049948434432, 11592835043328, 3548827054080, 760462940160, 108637562880, 9311791104, 362797056;

The row polynomials satisfy P(n,x) = (14+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 168*x + 36*x^2.

P[0, x_] := 196 + 168*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 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] = 196 + 168*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 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-37fb8ca4839f71ce, e-b35ead6daa50b714
• 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-37fb8ca4839f71ce, e-b35ead6daa50b714
• 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-37fb8ca4839f71ce, e-b35ead6daa50b714

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-37fb8ca4839f71ce

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-4ef268212a4f2f46 root: theorem check

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

    • parent_provers periodic_product e-4fdd1b3030c1c25c root: theorem check

      Parent-row provers of polytool polytool ?

• basis_map A013610 e-b35ead6daa50b714

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ca869591900be157 root: theorem check

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

    • parent_provers periodic_product e-48411a1ad7cd0ada 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.

• A038335 → basis A007318
• A038255 → 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