A128813: Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).

OEIS matches ?

• A128814: 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 ?
• A128813: 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 ?

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,   1;
  2: 1,   4,     3;
  3: 1,  10,    27,     18;
  4: 1,  20,   127,    288,     180;
  5: 1,  35,   427,   2193,    4500,      2700;
  6: 1,  56,  1162,  11160,   50553,     97200,     56700;
  7: 1,  84,  2730,  43696,  363033,   1512684,   2778300,    1587600;
  8: 1, 120,  5754, 141976, 1936089,  14581872,  57234924,  101606400,   57153600;
  9: 1, 165, 11154, 400906, 8325009, 101705877, 713419164, 2677177980, 4629441600, 2571912000;

The row polynomials satisfy P(n,x) = (1+((1/2)*n^2+(1/2)*n)*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 + ((1/2)*n^2 + (1/2)*n)*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 + ((1/2)*n^2 + (1/2)*n)*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,   1;
  2: 1,   4,     3;
  3: 1,  10,    27,     18;
  4: 1,  20,   127,    288,     180;
  5: 1,  35,   427,   2193,    4500,      2700;
  6: 1,  56,  1162,  11160,   50553,     97200,     56700;
  7: 1,  84,  2730,  43696,  363033,   1512684,   2778300,    1587600;
  8: 1, 120,  5754, 141976, 1936089,  14581872,  57234924,  101606400,   57153600;
  9: 1, 165, 11154, 400906, 8325009, 101705877, 713419164, 2677177980, 4629441600, 2571912000;

The row polynomials satisfy P(n,x) = (1+((1/2)*n^2+(1/2)*n)*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 + ((1/2)*n^2 + (1/2)*n)*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 + ((1/2)*n^2 + (1/2)*n)*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-fd268ef2c64764c5
• 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-a464fe87e26690eb, e-fd268ef2c64764c5
• 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-a464fe87e26690eb, e-fd268ef2c64764c5
• 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-a464fe87e26690eb, e-fd268ef2c64764c5
• 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-a464fe87e26690eb

Proof chains

2 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

• parent_provers paper_theorem e-a464fe87e26690eb root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-fd268ef2c64764c5 root: theorem check

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

Used in 7 proof steps

• e-201bedfe8f6594c1 basis_map (A013610) → basis_map_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
• e-42bb566932c0b2ac basis_map (A013609) → basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
• e-7d5e908b514e0d51 basis_map (A013611) → basis_map_A013611 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
• e-8bd02eafb0f0a53d basis_map (A007318) → basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
• e-cebfd0c72bfddbbf basis_map (A008277) → basis_map_A008277 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
• e-4ea38c3ef421f3f7 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
• e-4ea38c3ef421f3f7 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).

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