A062264: Coefficient triangle of certain polynomials N(4; m,x).

OEIS matches ?

• A002694: 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 ?
• A062264: 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,   5;
  2: 1,  12,   15;
  3: 1,  21,   63,    35;
  4: 1,  32,  168,   224,    70;
  5: 1,  45,  360,   840,   630,    126;
  6: 1,  60,  675,  2400,  3150,   1512,    210;
  7: 1,  77, 1155,  5775, 11550,   9702,   3234,   330;
  8: 1,  96, 1848, 12320, 34650,  44352,  25872,  6336,   495;
  9: 1, 117, 2808, 24024, 90090, 162162, 144144, 61776, 11583, 715;

The row polynomials satisfy (n^2+4*n)*P(n,x) = (n^2+4*n+(3*n^2+10*n+12)*x)*P(n-1,x) + ((2*n+4)*x-(2*n+4)*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 + 4*n + (3*n^2 + 10*n + 12)*x)*P[n - 1, x] + ((2*n + 4)*x - (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*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 + 4*n + (3*n^2 + 10*n + 12)*x)*v[i - 1] + ((2*n + 4)*x - (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*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,   5;
  2: 1,  12,   15;
  3: 1,  21,   63,    35;
  4: 1,  32,  168,   224,    70;
  5: 1,  45,  360,   840,   630,    126;
  6: 1,  60,  675,  2400,  3150,   1512,    210;
  7: 1,  77, 1155,  5775, 11550,   9702,   3234,   330;
  8: 1,  96, 1848, 12320, 34650,  44352,  25872,  6336,   495;
  9: 1, 117, 2808, 24024, 90090, 162162, 144144, 61776, 11583, 715;

The row polynomials satisfy (n^2+4*n)*P(n,x) = (n^2+4*n+(3*n^2+10*n+12)*x)*P(n-1,x) + ((2*n+4)*x-(2*n+4)*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 + 4*n + (3*n^2 + 10*n + 12)*x)*P[n - 1, x] + ((2*n + 4)*x - (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*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 + 4*n + (3*n^2 + 10*n + 12)*x)*v[i - 1] + ((2*n + 4)*x - (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*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-c22f7d753efa18e6
• 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-c22f7d753efa18e6
• 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-c22f7d753efa18e6
• 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-c22f7d753efa18e6

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-c22f7d753efa18e6 root: theorem check

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

Used in 7 proof steps

• e-06bdf5b2aa1ff3d5 preserver_image (affine, shift 0) → shift_x_plus_one of Coefficient triangle of certain polynomials N(4; m,x).
• e-3c18fb3613f81561 preserver_image (derivative_polynomial, shift 0) → derivative of Coefficient triangle of certain polynomials N(4; m,x).
• e-4f5eaf3b19bd0a90 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Coefficient triangle of certain polynomials N(4; m,x).
• e-53c50964658e5225 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Coefficient triangle of certain polynomials N(4; m,x).
• e-6d0521392d648070 preserver_image (reversal, shift 0) → reversal of Coefficient triangle of certain polynomials N(4; m,x).
• e-cfe6f0bc80bb6e33 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Coefficient triangle of certain polynomials N(4; m,x).
• e-e09e0bff614306ec preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Coefficient triangle of certain polynomials N(4; m,x).

Related sequences ?

Same fingerprint (1)

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

• A062264 → 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