A038319 → 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:            256,             320,             100;
  1:           4096,            7680,            4800,            1000;
  2:          65536,          163840,          153600,           64000,           10000;
  3:        1048576,         3276800,         4096000,         2560000,          800000,          100000;
  4:       16777216,        62914560,        98304000,        81920000,        38400000,         9600000,         1000000;
  5:      268435456,      1174405120,      2202009600,      2293760000,      1433600000,       537600000,       112000000,        10000000;
  6:     4294967296,     21474836480,     46976204800,     58720256000,     45875200000,     22937600000,      7168000000,      1280000000,      100000000;
  7:    68719476736,    386547056640,    966367641600,   1409286144000,   1321205760000,    825753600000,    344064000000,     92160000000,    14400000000,     1000000000;
  8:  1099511627776,   6871947673600,  19327352832000,  32212254720000,  35232153600000,  26424115200000,  13762560000000,   4915200000000,  1152000000000,   160000000000,   10000000000;
  9: 17592186044416, 120946279055360, 377957122048000, 708669603840000, 885837004800000, 775107379200000, 484442112000000, 216268800000000, 67584000000000, 14080000000000, 1760000000000, 100000000000;

The row polynomials satisfy P(n,x) = (16+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 320*x + 100*x^2.

P[0, x_] := 256 + 320*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 256 + 320*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 10*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-772e88d8a50e6734
• 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-772e88d8a50e6734
• 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-772e88d8a50e6734

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-772e88d8a50e6734

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f6c78a0c1a54b2b2 root: theorem check

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

    • parent_provers periodic_product e-97cca34dfbf5cc39 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.

• A038334 → basis A013611
• 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