A038271: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.

OEIS matches ?

• A013626: reversed triangle, 35 terms agree, 7 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A038271: 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 ?
• A001021: 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:        7,         5;
  2:       49,        70,        25;
  3:      343,       735,       525,        125;
  4:     2401,      6860,      7350,       3500,        625;
  5:    16807,     60025,     85750,      61250,      21875,      3125;
  6:   117649,    504210,    900375,     857500,     459375,    131250,     15625;
  7:   823543,   4117715,   8823675,   10504375,    7503125,   3215625,    765625,     78125;
  8:  5764801,  32941720,  82354300,  117649000,  105043750,  60025000,  21437500,   4375000,   390625;
  9: 40353607, 259416045, 741188700, 1235314500, 1323551250, 945393750, 450187500, 137812500, 24609375, 1953125;

The row polynomials satisfy P(n,x) = (7+5*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 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (7 + 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)

Triangle begins:
  0:       1;
  1:       5,        7;
  2:      25,       70,        49;
  3:     125,      525,       735,       343;
  4:     625,     3500,      7350,      6860,      2401;
  5:    3125,    21875,     61250,     85750,     60025,      16807;
  6:   15625,   131250,    459375,    857500,    900375,     504210,     117649;
  7:   78125,   765625,   3215625,   7503125,  10504375,    8823675,    4117715,    823543;
  8:  390625,  4375000,  21437500,  60025000, 105043750,  117649000,   82354300,  32941720,   5764801;
  9: 1953125, 24609375, 137812500, 450187500, 945393750, 1323551250, 1235314500, 741188700, 259416045, 40353607;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+5*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 + 5*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 + 5*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)

Triangle begins:
  0:        1;
  1:        7,         5;
  2:       49,        70,        25;
  3:      343,       735,       525,        125;
  4:     2401,      6860,      7350,       3500,        625;
  5:    16807,     60025,     85750,      61250,      21875,      3125;
  6:   117649,    504210,    900375,     857500,     459375,    131250,     15625;
  7:   823543,   4117715,   8823675,   10504375,    7503125,   3215625,    765625,     78125;
  8:  5764801,  32941720,  82354300,  117649000,  105043750,  60025000,  21437500,   4375000,   390625;
  9: 40353607, 259416045, 741188700, 1235314500, 1323551250, 945393750, 450187500, 137812500, 24609375, 1953125;

The row polynomials satisfy P(n,x) = (7+5*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 + 5*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 + 5*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-8a6d49b40ba3d6fa
• 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-8a6d49b40ba3d6fa, e-cb24020f7808d9ac
• 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-8a6d49b40ba3d6fa, e-cb24020f7808d9ac
• 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-8a6d49b40ba3d6fa
• 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-cb24020f7808d9ac
• 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-913c57e66973f22d
• 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-913c57e66973f22d

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-913c57e66973f22d root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-cb24020f7808d9ac root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-8a6d49b40ba3d6fa root: theorem check

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

Used in 7 proof steps

• e-5070060140385b53 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• e-62a1b0a86f60f435 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• e-7ab50ded9e353f77 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• e-cc1debec4a411a9c basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• e-e169cb33ba7b2c8c basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• e-eee83bb3f7c38b3e obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• e-eee83bb3f7c38b3e obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.

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