Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.

OEIS matches ?

• A001025: 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,           110,            25;
  1:         1331,          1815,           825,           125;
  2:        14641,         26620,         18150,          5500,           625;
  3:       161051,        366025,        332750,        151250,         34375,          3125;
  4:      1771561,       4831530,       5490375,       3327500,       1134375,        206250,         15625;
  5:     19487171,      62004635,      84551775,      64054375,      29115625,       7940625,       1203125,        78125;
  6:    214358881,     779486840,    1240092700,    1127357000,     640543750,     232925000,      52937500,      6875000,      390625;
  7:   2357947691,    9646149645,   17538453900,   18601390500,   12682766250,    5764893750,    1746937500,    340312500,    38671875,     1953125;
  8:  25937424601,  117897384550,  241153741125,  292307565000,  232517381250,  126827662500,   48040781250,  12478125000,  2126953125,   214843750,    9765625;
  9: 285311670611, 1426558353055, 3242178075125, 4421151920625, 4019229018750, 2557691193750, 1162586906250, 377463281250, 85787109375, 12998046875, 1181640625, 48828125;

The row polynomials satisfy P(n,x) = (11+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 110*x + 25*x^2.

P[0, x_] := 121 + 110*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 5*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 + 110*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 5*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-f6c78a0c1a54b2b2
• 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-97cca34dfbf5cc39, e-f6c78a0c1a54b2b2
• 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-97cca34dfbf5cc39, e-f6c78a0c1a54b2b2
• 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-f6c78a0c1a54b2b2
• 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-97cca34dfbf5cc39
• 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-34cbfee8ace30e91
• 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-34cbfee8ace30e91

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-34cbfee8ace30e91 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-97cca34dfbf5cc39 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-f6c78a0c1a54b2b2 root: theorem check

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

Used in 7 proof steps

• e-772e88d8a50e6734 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• e-82ec49a96db4a268 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• e-eb35900dd666694f basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• e-fd6222a072627022 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• e-fd91849598d32af1 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• e-7723a7442c6c45f3 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• e-7723a7442c6c45f3 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.

Related sequences ?

Same fingerprint (3)

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

• A038336 → basis A007318
• A038301 → basis A013611
• A038259 → basis A007318

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