Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.

OEIS matches ?

• A001026: 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:          121,           132,            36;
  1:         1331,          2178,          1188,           216;
  2:        14641,         31944,         26136,          9504,          1296;
  3:       161051,        439230,        479160,        261360,         71280,          7776;
  4:      1771561,       5797836,       7906140,       5749920,       2352240,        513216,         46656;
  5:     19487171,      74405562,     121754556,     110685960,      60374160,      19758816,       3592512,        279936;
  6:    214358881,     935384208,    1785733488,    1948072896,    1328231520,     579591936,     158070528,      24634368,      1679616;
  7:   2357947691,   11575379574,   25255373616,   32143202784,   26298984096,   14344900416,    5216327424,    1219401216,    166281984,    10077696;
  8:  25937424601,  141476861460,  347261387220,  505107472320,  482148041760,  315587809152,  143449004160,   44711377920,   9145509120,  1108546560,   60466176;
  9: 285311670611, 1711870023666, 4668736428180, 7639750518840, 8334273293280, 6364354151232, 3471465900672, 1352519182080, 368868867840, 67067066880, 7316407296, 362797056;

The row polynomials satisfy P(n,x) = (11+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 132*x + 36*x^2.

P[0, x_] := 121 + 132*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 121 + 132*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 6*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-940e6d9217c9d73c
• 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-2fddb004b6f97d1c, e-940e6d9217c9d73c, e-de8595a50a8969d6
• 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-2fddb004b6f97d1c, e-940e6d9217c9d73c, e-de8595a50a8969d6
• 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-940e6d9217c9d73c
• 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-2fddb004b6f97d1c, e-de8595a50a8969d6
• 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-45f78ad8817d1350
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 11] to roots in (-inf, 0]: an input root r <= 11 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-45f78ad8817d1350

Proof chains

4 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 A013609 e-2fddb004b6f97d1c

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-da334b7c58401d78 root: theorem check

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

    • parent_provers periodic_product e-c5656743efdffc45 root: theorem check

      Parent-row provers of polytool polytool ?

• basis_map_theorem e-45f78ad8817d1350 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-de8595a50a8969d6 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-940e6d9217c9d73c root: theorem check

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

Used in 7 proof steps

• e-572c9115d911d6cf basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• e-5fcdd9bd1fb647dd basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• e-72f5203b9bd4f955 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• e-7304be3f66cc3caf basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• e-a19af80736e38760 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• e-a06c6022a905308b obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• e-a06c6022a905308b obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.

Related sequences ?

Same fingerprint (3)

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

• A038314 → basis A007318
• A038277 → basis A013610
• A013618 → basis A013609

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