A038334 → 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:             400,             1280,              1024;
  1:            8000,            38400,             61440,              32768;
  2:          160000,          1024000,           2457600,            2621440,            1048576;
  3:         3200000,         25600000,          81920000,          131072000,          104857600,           33554432;
  4:        64000000,        614400000,        2457600000,         5242880000,         6291456000,         4026531840,          1073741824;
  5:      1280000000,      14336000000,       68812800000,       183500800000,       293601280000,       281857228800,        150323855360,         34359738368;
  6:     25600000000,     327680000000,     1835008000000,      5872025600000,     11744051200000,     15032385536000,      12025908428800,       5497558138880,       1099511627776;
  7:    512000000000,    7372800000000,    47185920000000,    176160768000000,    422785843200000,    676457349120000,     721554505728000,     494780232499200,     197912092999680,     35184372088832;
  8:  10240000000000,  163840000000000,  1179648000000000,   5033164800000000,  14092861440000000,  27058293964800000,   36077725286400000,   32985348833280000,   19791209299968000,   7036874417766400,   1125899906842624;
  9: 204800000000000, 3604480000000000, 28835840000000000, 138412032000000000, 442918502400000000, 992137445376000000, 1587419912601600000, 1814194185830400000, 1451355348664320000, 774056185954304000, 247697979505377280, 36028797018963968;

The row polynomials satisfy P(n,x) = (20+32*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1280*x + 1024*x^2.

P[0, x_] := 400 + 1280*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 32*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1280*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 32*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-bd5ba1e8beeebdaa
• 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-bd5ba1e8beeebdaa
• 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-bd5ba1e8beeebdaa

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-bd5ba1e8beeebdaa

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f4f0736bd14a35db root: theorem check

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

    • parent_provers periodic_product e-b445a5ff4f115ce0 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.

• A038319 → basis A013609
• A038220 → basis A013611
• A013622 → 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