A013613: Triangle of coefficients in expansion of (1+6x)^n.

OEIS matches ?

• A000420: 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 ?
• A013613: 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 ?
• A038255: reversed triangle, 43 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 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: 1,  6;
  2: 1, 12,   36;
  3: 1, 18,  108,   216;
  4: 1, 24,  216,   864,   1296;
  5: 1, 30,  360,  2160,   6480,   7776;
  6: 1, 36,  540,  4320,  19440,  46656,   46656;
  7: 1, 42,  756,  7560,  45360, 163296,  326592,   279936;
  8: 1, 48, 1008, 12096,  90720, 435456, 1306368,  2239488,  1679616;
  9: 1, 54, 1296, 18144, 163296, 979776, 3919104, 10077696, 15116544, 10077696;

The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 6*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] = (1 + 6*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: 1,  6;
  2: 1, 12,   36;
  3: 1, 18,  108,   216;
  4: 1, 24,  216,   864,   1296;
  5: 1, 30,  360,  2160,   6480,   7776;
  6: 1, 36,  540,  4320,  19440,  46656,   46656;
  7: 1, 42,  756,  7560,  45360, 163296,  326592,   279936;
  8: 1, 48, 1008, 12096,  90720, 435456, 1306368,  2239488,  1679616;
  9: 1, 54, 1296, 18144, 163296, 979776, 3919104, 10077696, 15116544, 10077696;

The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 6*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] = (1 + 6*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)

Triangle begins:
  0:        1;
  1:        6,        1;
  2:       36,       12,        1;
  3:      216,      108,       18,       1;
  4:     1296,      864,      216,      24,      1;
  5:     7776,     6480,     2160,     360,     30,      1;
  6:    46656,    46656,    19440,    4320,    540,     36,     1;
  7:   279936,   326592,   163296,   45360,   7560,    756,    42,    1;
  8:  1679616,  2239488,  1306368,  435456,  90720,  12096,  1008,   48,  1;
  9: 10077696, 15116544, 10077696, 3919104, 979776, 163296, 18144, 1296, 54, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 6*x)*P[n - 1, x]];
Table[Reverse[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] = (1 + 6*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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-1784cac1b85d5f84
• 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-1784cac1b85d5f84, e-5658c7be1cbffd3f
• 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-1784cac1b85d5f84, e-5658c7be1cbffd3f
• 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-1784cac1b85d5f84
• 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-5658c7be1cbffd3f
• 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-79894ccac0604a0f
• 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-79894ccac0604a0f

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-79894ccac0604a0f root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-5658c7be1cbffd3f root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-1784cac1b85d5f84 root: theorem check

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

Used in 7 proof steps

• e-64a82ef46e481e9c basis_map (A013609) → basis_map_A013609 of Triangle of coefficients in expansion of (1+6x)^n.
• e-694953561e6f80c1 basis_map (A013610) → basis_map_A013610 of Triangle of coefficients in expansion of (1+6x)^n.
• e-7bcf400b3d75a9d5 basis_map (A008277) → basis_map_A008277 of Triangle of coefficients in expansion of (1+6x)^n.
• e-83999c64158b130e basis_map (A013611) → basis_map_A013611 of Triangle of coefficients in expansion of (1+6x)^n.
• e-c08496e978bf378e basis_map (A007318) → basis_map_A007318 of Triangle of coefficients in expansion of (1+6x)^n.
• e-84f0fe7fd2f0b117 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (1+6x)^n.
• e-84f0fe7fd2f0b117 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (1+6x)^n.

Related sequences ?

Same fingerprint (2)

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

• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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