A062190: Coefficient triangle of certain polynomials N(5; m,x).

OEIS matches ?

• A003516: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 2 ?
• A062190: triangle, 55 terms agree, 10 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,  14,   21;
  3: 1,  24,   84,    56;
  4: 1,  36,  216,   336,    126;
  5: 1,  50,  450,  1200,   1050,    252;
  6: 1,  66,  825,  3300,   4950,   2772,    462;
  7: 1,  84, 1386,  7700,  17325,  16632,   6468,    792;
  8: 1, 104, 2184, 16016,  50050,  72072,  48048,  13728,  1287;
  9: 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002;

The row polynomials satisfy (n^2+5*n)*P(n,x) = (n^2+5*n+(3*n^2+13*n+20)*x)*P(n-1,x) + ((2*n+5)*x-(2*n+5)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + (3*n^2 + 13*n + 20)*x)*P[n - 1, x] + ((2*n + 5)*x - (2*n + 5)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n)]];
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] = ((n^2 + 5*n + (3*n^2 + 13*n + 20)*x)*v[i - 1] + ((2*n + 5)*x - (2*n + 5)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n)); 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,  14,   21;
  3: 1,  24,   84,    56;
  4: 1,  36,  216,   336,    126;
  5: 1,  50,  450,  1200,   1050,    252;
  6: 1,  66,  825,  3300,   4950,   2772,    462;
  7: 1,  84, 1386,  7700,  17325,  16632,   6468,    792;
  8: 1, 104, 2184, 16016,  50050,  72072,  48048,  13728,  1287;
  9: 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002;

The row polynomials satisfy (n^2+5*n)*P(n,x) = (n^2+5*n+(3*n^2+13*n+20)*x)*P(n-1,x) + ((2*n+5)*x-(2*n+5)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + (3*n^2 + 13*n + 20)*x)*P[n - 1, x] + ((2*n + 5)*x - (2*n + 5)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n)]];
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] = ((n^2 + 5*n + (3*n^2 + 13*n + 20)*x)*v[i - 1] + ((2*n + 5)*x - (2*n + 5)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n)); 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-e968c429d4b85e0b
• 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-e968c429d4b85e0b
• 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-e968c429d4b85e0b
• 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-e968c429d4b85e0b

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• theorem_check liu_wang e-e968c429d4b85e0b root: theorem check

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

Used in 6 proof steps

• e-581332e7358bcae7 preserver_image (derivative_polynomial, shift 0) → derivative of Coefficient triangle of certain polynomials N(5; m,x).
• e-595b0519ab8ba235 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Coefficient triangle of certain polynomials N(5; m,x).
• e-725543b0eb30a334 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Coefficient triangle of certain polynomials N(5; m,x).
• e-888afb9e2aff2a08 preserver_image (reversal, shift 0) → reversal of Coefficient triangle of certain polynomials N(5; m,x).
• e-e91167a9e27d414d preserver_image (affine, shift 0) → shift_x_plus_one of Coefficient triangle of certain polynomials N(5; m,x).
• e-f9926095268ea913 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Coefficient triangle of certain polynomials N(5; m,x).

Related sequences ?

Same fingerprint (1)

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

• A062190 → reverse

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