A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.

OEIS matches ?

• A001027: 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 ?
• A038277: triangle, 32 terms agree, 7 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A038321: reversed triangle, 36 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:        7,        11;
  2:       49,       154,        121;
  3:      343,      1617,       2541,        1331;
  4:     2401,     15092,      35574,       37268,       14641;
  5:    16807,    132055,     415030,      652190,      512435,      161051;
  6:   117649,   1109262,    4357815,     9130660,    10761135,     6764142,     1771561;
  7:   823543,   9058973,   42706587,   111850585,   175765205,   165721479,    86806489,    19487171;
  8:  5764801,  72471784,  398594812,  1252726552,  2460712870,  3093467608,  2430581692,  1091281576,   214358881;
  9: 40353607, 570715299, 3587353308, 13153628796, 31004982162, 48722114826, 51042215532, 34375369644, 13504609503, 2357947691;

The row polynomials satisfy P(n,x) = (7+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(7 + 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; for(n = 1, N, my(i = n + 1); v[i] = (7 + 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)

Triangle begins:
  0:        1;
  1:        7,        11;
  2:       49,       154,        121;
  3:      343,      1617,       2541,        1331;
  4:     2401,     15092,      35574,       37268,       14641;
  5:    16807,    132055,     415030,      652190,      512435,      161051;
  6:   117649,   1109262,    4357815,     9130660,    10761135,     6764142,     1771561;
  7:   823543,   9058973,   42706587,   111850585,   175765205,   165721479,    86806489,    19487171;
  8:  5764801,  72471784,  398594812,  1252726552,  2460712870,  3093467608,  2430581692,  1091281576,   214358881;
  9: 40353607, 570715299, 3587353308, 13153628796, 31004982162, 48722114826, 51042215532, 34375369644, 13504609503, 2357947691;

The row polynomials satisfy P(n,x) = (7+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(7 + 11*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] = (7 + 11*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:         11,           7;
  2:        121,         154,          49;
  3:       1331,        2541,        1617,         343;
  4:      14641,       37268,       35574,       15092,        2401;
  5:     161051,      512435,      652190,      415030,      132055,       16807;
  6:    1771561,     6764142,    10761135,     9130660,     4357815,     1109262,      117649;
  7:   19487171,    86806489,   165721479,   175765205,   111850585,    42706587,     9058973,     823543;
  8:  214358881,  1091281576,  2430581692,  3093467608,  2460712870,  1252726552,   398594812,   72471784,   5764801;
  9: 2357947691, 13504609503, 34375369644, 51042215532, 48722114826, 31004982162, 13153628796, 3587353308, 570715299, 40353607;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(7 + 11*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] = (7 + 11*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-f9e3d73ad0eb8fa6
• 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-399c4cc80f25ee4f, e-f9e3d73ad0eb8fa6
• 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-399c4cc80f25ee4f, e-f9e3d73ad0eb8fa6
• 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-f9e3d73ad0eb8fa6
• 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-399c4cc80f25ee4f
• 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-dc6cd425a1850984
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 7] to roots in (-inf, 0]: an input root r <= 7 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-dc6cd425a1850984

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-dc6cd425a1850984 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-399c4cc80f25ee4f root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-f9e3d73ad0eb8fa6 root: theorem check

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

Used in 7 proof steps

• e-59f75239fe41171d basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
• e-77c9a747c1207e17 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
• e-8cea3676de7ce3cf basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
• e-c73faa2b910a081c basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
• e-f0378e56b9f5156d basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
• e-3667cd139473d050 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
• e-3667cd139473d050 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.

Related sequences ?

Same fingerprint (1)

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

• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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