A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.

OEIS matches ?

• A001020: 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:         64,          48,           9;
  1:        512,         576,         216,          27;
  2:       4096,        6144,        3456,         864,          81;
  3:      32768,       61440,       46080,       17280,        3240,         243;
  4:     262144,      589824,      552960,      276480,       77760,       11664,         729;
  5:    2097152,     5505024,     6193152,     3870720,     1451520,      326592,       40824,       2187;
  6:   16777216,    50331648,    66060288,    49545216,    23224320,     6967296,     1306368,     139968,      6561;
  7:  134217728,   452984832,   679477248,   594542592,   334430208,   125411328,    31352832,    5038848,    472392,    19683;
  8: 1073741824,  4026531840,  6794772480,  6794772480,  4459069440,  2006581248,   627056640,  134369280,  18895680,  1574640,   59049;
  9: 8589934592, 35433480192, 66437775360, 74742497280, 56056872960, 29429858304, 11036196864, 2956124160, 554273280, 69284160, 5196312, 177147;

The row polynomials satisfy P(n,x) = (8+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 48*x + 9*x^2.

P[0, x_] := 64 + 48*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 64 + 48*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + 3*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-da334b7c58401d78
• 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-c5656743efdffc45, e-da334b7c58401d78
• 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-c5656743efdffc45, e-da334b7c58401d78
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-da334b7c58401d78
• 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-c5656743efdffc45
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-a1af237376376fe3
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 8] to roots in (-inf, 0]: an input root r <= 8 gives the root (r - a)/d <= 0; a larger root gives a positive root
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-a1af237376376fe3

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map_theorem e-a1af237376376fe3 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-c5656743efdffc45 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-da334b7c58401d78 root: theorem check

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

Used in 7 proof steps

• e-2fddb004b6f97d1c basis_map (A013609) → Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• e-8f18b93eb5dfe7b3 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• e-d7f363249f6694ec basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• e-e643036831fc9f95 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• e-f67f09ded1c0d580 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• e-b64329fdd6480cca obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• e-b64329fdd6480cca obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.

Related sequences ?

Same fingerprint (4)

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

• A120909 → basis A013611
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• A013609 → basis A013611
• A027466 → 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