Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.

OEIS matches ?

• A001022: 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:          100,           60,            9;
  1:         1000,          900,          270,           27;
  2:        10000,        12000,         5400,         1080,           81;
  3:       100000,       150000,        90000,        27000,         4050,          243;
  4:      1000000,      1800000,      1350000,       540000,       121500,        14580,         729;
  5:     10000000,     21000000,     18900000,      9450000,      2835000,       510300,       51030,       2187;
  6:    100000000,    240000000,    252000000,    151200000,     56700000,     13608000,     2041200,     174960,       6561;
  7:   1000000000,   2700000000,   3240000000,   2268000000,   1020600000,    306180000,    61236000,    7873200,     590490,     19683;
  8:  10000000000,  30000000000,  40500000000,  32400000000,  17010000000,   6123600000,  1530900000,  262440000,   29524500,   1968300,   59049;
  9: 100000000000, 330000000000, 495000000000, 445500000000, 267300000000, 112266000000, 33679800000, 7217100000, 1082565000, 108256500, 6495390, 177147;

The row polynomials satisfy P(n,x) = (10+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 60*x + 9*x^2.

P[0, x_] := 100 + 60*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(10 + 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] = 100 + 60*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + 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-e186153c671b61f0
• 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-e186153c671b61f0, e-f733aee5869de636
• 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-e186153c671b61f0, e-f733aee5869de636
• 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-e186153c671b61f0
• 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-f733aee5869de636
• 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-7687994fc4fefbbe
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 10] to roots in (-inf, 0]: an input root r <= 10 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-7687994fc4fefbbe

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-7687994fc4fefbbe root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-f733aee5869de636 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-e186153c671b61f0 root: theorem check

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

Used in 7 proof steps

• e-1eab37781c60e76a basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• e-54bb49968ded72d0 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• e-7b93be005ec5b750 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• e-c80be26a429cc3fc basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• e-ebc4517445c15609 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• e-e4e37e9713aaf48a obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• e-e4e37e9713aaf48a obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.

Related sequences ?

Same fingerprint (2)

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

• A038291 → basis A013610
• A013612 → 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