A038328 → basis A013611

OEIS matches ?

• A009966: 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,            224,             64;
  1:          2744,           4704,           2688,             512;
  2:         38416,          87808,          75264,           28672,            4096;
  3:        537824,        1536640,        1756160,         1003520,          286720,           32768;
  4:       7529536,       25815552,       36879360,        28098560,        12042240,         2752512,         262144;
  5:     105413504,      421654016,      722835456,       688414720,       393379840,       134873088,       25690112,        2097152;
  6:    1475789056,     6746464256,    13492928512,     15420489728,     11014635520,      5035261952,     1438646272,      234881024,      16777216;
  7:   20661046784,   106256812032,   242872713216,    323830284288,    277568815104,    158610751488,    60423143424,    14797504512,    2113929216,     134217728;
  8:  289254654976,  1652883742720,  4250272481280,   6476605685760,   6476605685760,   4441101041664,  2114810019840,   690550210560,  147975045120,   18790481920,   1073741824;
  9: 4049565169664, 25454409637888, 72726884679680, 124674659450880, 142485325086720, 113988260069376, 65136148611072, 26586183106560, 7596052316160, 1446867107840, 165356240896, 8589934592;

The row polynomials satisfy P(n,x) = (14+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 224*x + 64*x^2.

P[0, x_] := 196 + 224*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 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] = 196 + 224*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 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-0eb3975e7d8d74fc
• 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-0eb3975e7d8d74fc
• 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-0eb3975e7d8d74fc

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-0eb3975e7d8d74fc

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ca869591900be157 root: theorem check

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

    • parent_provers periodic_product e-48411a1ad7cd0ada root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A038255 → basis A013611

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