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

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

P[0, x_] := 9 + 48*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 9 + 48*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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-fdf6b848b46b3343
• 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-711ca5eff5ba56ea, e-fdf6b848b46b3343
• 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-711ca5eff5ba56ea, e-fdf6b848b46b3343
• 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-fdf6b848b46b3343
• 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-711ca5eff5ba56ea
• 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-71cb1ba62d8762b5
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 3] to roots in (-inf, 0]: an input root r <= 3 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-71cb1ba62d8762b5

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-71cb1ba62d8762b5 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-711ca5eff5ba56ea root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-fdf6b848b46b3343 root: theorem check

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

Used in 7 proof steps

• e-0566c972c7be0ca0 basis_map (A007318) → basis_map_A007318 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• e-376f2b9433b6ff6b basis_map (A013611) → basis_map_A013611 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• e-c8a2ee9cf24a53ed basis_map (A013609) → basis_map_A013609 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• e-edadd00b11ba8928 basis_map (A013610) → basis_map_A013610 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• e-fc0676c2630bb222 basis_map (A008277) → basis_map_A008277 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• e-c18c7d7bde35dd59 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)…
• e-c18c7d7bde35dd59 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)…

Related sequences ?

Same fingerprint (4)

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

• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• A013609 → basis A013611
• A027466 → basis A013610
• A120909 → 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