A038284 → basis A013609

OEIS matches ?

• A009970: 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,            336,             144;
  1:          2744,           7056,            6048,            1728;
  2:         38416,         131712,          169344,           96768,           20736;
  3:        537824,        2304960,         3951360,         3386880,         1451520,          248832;
  4:       7529536,       38723328,        82978560,        94832640,        60963840,        20901888,         2985984;
  5:     105413504,      632481024,      1626379776,      2323399680,      1991485440,      1024192512,       292626432,        35831808;
  6:    1475789056,    10119696384,     30359089152,     52044152832,     55761592320,     38236520448,     16387080192,      4013162496,       429981696;
  7:   20661046784,   159385218048,    546463604736,   1092927209472,   1405192126464,   1204450394112,    688257368064,    252829237248,     54177693696,     5159780352;
  8:  289254654976,  2479325614080,   9563113082880,  21858544189440,  32787816284160,  33724611035136,  24089007882240,  11798697738240,   3792438558720,   722369249280,   61917364224;
  9: 4049565169664, 38181614456832, 163635490529280, 420776975646720, 721331958251520, 865598349901824, 741941442772992, 454249862922240, 194678512680960, 55622432194560, 9535274090496, 743008370688;

The row polynomials satisfy P(n,x) = (14+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 336*x + 144*x^2.

P[0, x_] := 196 + 336*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 12*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 + 336*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 12*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-65e466871b01ed5e
• 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-65e466871b01ed5e
• 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-65e466871b01ed5e

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-65e466871b01ed5e

  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 ?

Related sequences ?

Same fingerprint (3)

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

• A038321 → basis A013610
• A038335 → basis A013609
• A013613 → basis A007318

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