Triangle of coefficients in expansion of (1+5x)^n.

OEIS matches ?

• A000400: 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, 10,   25;
  1: 1, 15,   75,   125;
  2: 1, 20,  150,   500,    625;
  3: 1, 25,  250,  1250,   3125,    3125;
  4: 1, 30,  375,  2500,   9375,   18750,   15625;
  5: 1, 35,  525,  4375,  21875,   65625,  109375,    78125;
  6: 1, 40,  700,  7000,  43750,  175000,  437500,   625000,   390625;
  7: 1, 45,  900, 10500,  78750,  393750, 1312500,  2812500,  3515625,   1953125;
  8: 1, 50, 1125, 15000, 131250,  787500, 3281250,  9375000, 17578125,  19531250,   9765625;
  9: 1, 55, 1375, 20625, 206250, 1443750, 7218750, 25781250, 64453125, 107421875, 107421875, 48828125;

The row polynomials satisfy P(n,x) = (1+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 10*x + 25*x^2.

P[0, x_] := 1 + 10*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 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 + 10*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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)

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-8bdb70660e92b5b3
• 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-41a50f5538a6b055, e-8bdb70660e92b5b3
• 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-41a50f5538a6b055, e-8bdb70660e92b5b3
• 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-8bdb70660e92b5b3
• 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-41a50f5538a6b055
• 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-89b686e6b2113e81
• 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-89b686e6b2113e81

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-89b686e6b2113e81 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-41a50f5538a6b055 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-8bdb70660e92b5b3 root: theorem check

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

Used in 7 proof steps

• e-31086658f6c15e80 basis_map (A013609) → basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.
• e-46de8f4df4fd1cb3 basis_map (A013610) → basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.
• e-a1d582096966c77c basis_map (A007318) → basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.
• e-bab24e2df6037a14 basis_map (A008277) → basis_map_A008277 of Triangle of coefficients in expansion of (1+5x)^n.
• e-db3287eb465d1cf9 basis_map (A013611) → basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.
• e-0f5b357fc0453aeb obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (1+5x)^n.
• e-0f5b357fc0453aeb obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (1+5x)^n.

Related sequences ?

Same fingerprint (5)

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

• A038329 → basis A007318
• A038291 → basis A013609
• A147716 → basis A013610
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
• A013611 → x ↦ x/(x+1)

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