Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.

OEIS matches ?

• A001027: 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,           154,            49;
  1:         1331,          2541,          1617,            343;
  2:        14641,         37268,         35574,          15092,           2401;
  3:       161051,        512435,        652190,         415030,         132055,          16807;
  4:      1771561,       6764142,      10761135,        9130660,        4357815,        1109262,        117649;
  5:     19487171,      86806489,     165721479,      175765205,      111850585,       42706587,       9058973,        823543;
  6:    214358881,    1091281576,    2430581692,     3093467608,     2460712870,     1252726552,     398594812,      72471784,       5764801;
  7:   2357947691,   13504609503,   34375369644,    51042215532,    48722114826,    31004982162,   13153628796,    3587353308,     570715299,     40353607;
  8:  25937424601,  165056338370,  472661332605,   802091958360,   893238771810,   682109607564,  361724791890,  131536287960,   31389341445,   4438896770,   282475249;
  9: 285311670611, 1997181694277, 6354669027245, 12131640870195, 15440270198430, 13755877085874, 8753739963738, 3978972710790, 1266036771615, 268553254585, 34179505129, 1977326743;

The row polynomials satisfy P(n,x) = (11+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 154*x + 49*x^2.

P[0, x_] := 121 + 154*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 7*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 + 154*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 7*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-f6f8234ea600f6f8
• 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-16634379058ff4dd, e-f6f8234ea600f6f8
• 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-16634379058ff4dd, e-f6f8234ea600f6f8
• 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-f6f8234ea600f6f8
• 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-16634379058ff4dd
• 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-78bb8973195ccb8f
• 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-78bb8973195ccb8f

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-78bb8973195ccb8f root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-16634379058ff4dd root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-f6f8234ea600f6f8 root: theorem check

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

Used in 7 proof steps

• e-322a2e4c0bbd3231 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
• e-5a47242ff4b92549 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
• e-632cc4d34c6e419b basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
• e-86a79a465fd2222b basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
• e-e460f89b751823ed basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
• e-57fae4cb2b7c6f75 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
• e-57fae4cb2b7c6f75 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.

Related sequences ?

Same fingerprint (1)

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

• A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.

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