Triangle of coefficients in expansion of (1+11x)^n.

OEIS matches ?

• A001021: 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: 1,  22,  121;
  1: 1,  33,  363,   1331;
  2: 1,  44,  726,   5324,   14641;
  3: 1,  55, 1210,  13310,   73205,   161051;
  4: 1,  66, 1815,  26620,  219615,   966306,   1771561;
  5: 1,  77, 2541,  46585,  512435,  3382071,  12400927,   19487171;
  6: 1,  88, 3388,  74536, 1024870,  9018856,  49603708,  155897368,   214358881;
  7: 1,  99, 4356, 111804, 1844766, 20292426, 148811124,  701538156,  1929229929,   2357947691;
  8: 1, 110, 5445, 159720, 3074610, 40584852, 372027810, 2338460520,  9646149645,  23579476910,  25937424601;
  9: 1, 121, 6655, 219615, 4831530, 74405562, 818461182, 6430766430, 35369215365, 129687123005, 285311670611, 285311670611;

The row polynomials satisfy P(n,x) = (1+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 22*x + 121*x^2.

P[0, x_] := 1 + 22*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 11*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 22*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + 11*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-5d158048399db1f0
• 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-5d158048399db1f0, e-70681fa17f5604ff
• 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-5d158048399db1f0, e-70681fa17f5604ff
• 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-5d158048399db1f0
• 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-70681fa17f5604ff
• 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-565be378e09842fb
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 1] to roots in (-inf, 0]: an input root r <= 1 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-565be378e09842fb

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-565be378e09842fb root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-70681fa17f5604ff root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-5d158048399db1f0 root: theorem check

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

Used in 7 proof steps

• e-107a218257e6f127 basis_map (A007318) → Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
• e-4217afbeac9f09d9 basis_map (A013610) → basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.
• e-4aabe19e5745ac70 basis_map (A013609) → basis_map_A013609 of Triangle of coefficients in expansion of (1+11x)^n.
• e-e469bcf1fd1b08af basis_map (A008277) → basis_map_A008277 of Triangle of coefficients in expansion of (1+11x)^n.
• e-f283d143fa0950f0 basis_map (A013611) → basis_map_A013611 of Triangle of coefficients in expansion of (1+11x)^n.
• e-f27f365b1aa97cdc obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (1+11x)^n.
• e-f27f365b1aa97cdc obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (1+11x)^n.

Related sequences ?

Same fingerprint (1)

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

• A038303 → 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