A038334 → basis A007318

OEIS matches ?

• A009972: 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:             400,             320,               64;
  1:            8000,            9600,             3840,              512;
  2:          160000,          256000,           153600,            40960,             4096;
  3:         3200000,         6400000,          5120000,          2048000,           409600,           32768;
  4:        64000000,       153600000,        153600000,         81920000,         24576000,         3932160,          262144;
  5:      1280000000,      3584000000,       4300800000,       2867200000,       1146880000,       275251200,        36700160,         2097152;
  6:     25600000000,     81920000000,     114688000000,      91750400000,      45875200000,     14680064000,      2936012800,       335544320,       16777216;
  7:    512000000000,   1843200000000,    2949120000000,    2752512000000,    1651507200000,    660602880000,    176160768000,     30198988800,     3019898880,     134217728;
  8:  10240000000000,  40960000000000,   73728000000000,   78643200000000,   55050240000000,  26424115200000,   8808038400000,   2013265920000,   301989888000,   26843545600,   1073741824;
  9: 204800000000000, 901120000000000, 1802240000000000, 2162688000000000, 1730150400000000, 968884224000000, 387553689600000, 110729625600000, 22145925120000, 2952790016000, 236223201280, 8589934592;

The row polynomials satisfy P(n,x) = (20+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 320*x + 64*x^2.

P[0, x_] := 400 + 320*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 8*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 + 320*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 8*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-ec87a016e71e40fc
• 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-ec87a016e71e40fc
• 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-ec87a016e71e40fc

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 A007318 e-ec87a016e71e40fc

  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 (6)

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

• A038291 → basis A013611
• A013612 → basis A013610
• A038329 → basis A013609
• A038220 → basis A007318
• A013622 → basis A013611
• A013620 → x ↦ x/(x+1)

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