A147716: Triangle of coefficients in expansion of (14 + x)^n.

OEIS matches ?

• A147716: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A001024: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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;
  1:          14,           1;
  2:         196,          28,          1;
  3:        2744,         588,         42,         1;
  4:       38416,       10976,       1176,        56,        1;
  5:      537824,      192080,      27440,      1960,       70,       1;
  6:     7529536,     3226944,     576240,     54880,     2940,      84,      1;
  7:   105413504,    52706752,   11294304,   1344560,    96040,    4116,     98,    1;
  8:  1475789056,   843308032,  210827008,  30118144,  2689120,  153664,   5488,  112,   1;
  9: 20661046784, 13282101504, 3794886144, 632481024, 67765824, 4840416, 230496, 7056, 126, 1;

The row polynomials satisfy P(n,x) = (14+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(14 + 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; for(n = 1, N, my(i = n + 1); v[i] = (14 + 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)

Triangle begins:
  0:           1;
  1:          14,           1;
  2:         196,          28,          1;
  3:        2744,         588,         42,         1;
  4:       38416,       10976,       1176,        56,        1;
  5:      537824,      192080,      27440,      1960,       70,       1;
  6:     7529536,     3226944,     576240,     54880,     2940,      84,      1;
  7:   105413504,    52706752,   11294304,   1344560,    96040,    4116,     98,    1;
  8:  1475789056,   843308032,  210827008,  30118144,  2689120,  153664,   5488,  112,   1;
  9: 20661046784, 13282101504, 3794886144, 632481024, 67765824, 4840416, 230496, 7056, 126, 1;

The row polynomials satisfy P(n,x) = (14+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(14 + x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (14 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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-08b3ad2096e0d7df
• 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-08b3ad2096e0d7df, e-7403f1e58efb6a95
• 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-08b3ad2096e0d7df, e-7403f1e58efb6a95
• 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-08b3ad2096e0d7df
• 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-7403f1e58efb6a95
• 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-8e6ae20eb3f6019c
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 14] to roots in (-inf, 0]: an input root r <= 14 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-8e6ae20eb3f6019c

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-8e6ae20eb3f6019c root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-7403f1e58efb6a95 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-08b3ad2096e0d7df root: theorem check

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

Used in 7 proof steps

• e-011a72a323f5e83e basis_map (A013610) → basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.
• e-267357ec4c02d3cf basis_map (A013609) → basis_map_A013609 of Triangle of coefficients in expansion of (14 + x)^n.
• e-7eb5590197b773bf basis_map (A007318) → Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).
• e-8d978ffb07d0025e basis_map (A013611) → basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.
• e-9d8b8ca223112425 basis_map (A008277) → basis_map_A008277 of Triangle of coefficients in expansion of (14 + x)^n.
• e-82f056fc366df894 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (14 + x)^n.
• e-82f056fc366df894 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (14 + x)^n.

Related sequences ?

No candidate equalities or related sequences.

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