A038319 → basis A013611

OEIS matches ?

• A009980: 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,             640,              400;
  1:           4096,           15360,            19200,             8000;
  2:          65536,          327680,           614400,           512000,            160000;
  3:        1048576,         6553600,         16384000,         20480000,          12800000,           3200000;
  4:       16777216,       125829120,        393216000,        655360000,         614400000,         307200000,          64000000;
  5:      268435456,      2348810240,       8808038400,      18350080000,       22937600000,       17203200000,        7168000000,        1280000000;
  6:     4294967296,     42949672960,     187904819200,     469762048000,      734003200000,      734003200000,      458752000000,      163840000000,       25600000000;
  7:    68719476736,    773094113280,    3865470566400,   11274289152000,    21139292160000,    26424115200000,    22020096000000,    11796480000000,     3686400000000,     512000000000;
  8:  1099511627776,  13743895347200,   77309411328000,  257698037760000,   563714457600000,   845571686400000,   880803840000000,   629145600000000,   294912000000000,   81920000000000,   10240000000000;
  9: 17592186044416, 241892558110720, 1511828488192000, 5669356830720000, 14173392076800000, 24803436134400000, 31004295168000000, 27682406400000000, 17301504000000000, 7208960000000000, 1802240000000000, 204800000000000;

The row polynomials satisfy P(n,x) = (16+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 640*x + 400*x^2.

P[0, x_] := 256 + 640*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 20*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 + 640*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 20*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-fd91849598d32af1
• 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-fd91849598d32af1
• 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-fd91849598d32af1

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-fd91849598d32af1

  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 (6)

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

• A013622 → basis A013609
• A013611 → basis A007318
• A038329 → basis A013611
• A038231 → x ↦ x/(x+1)
• A038334 → basis A013609
• A038271 → 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