A121574 → reverse

OEIS matches ?

• A121574: reversed 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 ?
• A020704: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A006190: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,  2;
  2: 1,  5,   4;
  3: 1,  8,  16,    8;
  4: 1, 11,  37,   44,   16;
  5: 1, 14,  67,  134,  112,    32;
  6: 1, 17, 106,  305,  424,   272,    64;
  7: 1, 20, 154,  584, 1168,  1232,   640,  128;
  8: 1, 23, 211,  998, 2641,  3992,  3376, 1472,  256;
  9: 1, 26, 277, 1574, 5221, 10442, 12592, 8864, 3328, 512;

The row polynomials satisfy n*P(n,x) = (n+(3*n^2-n)*x-(4*n-4)*x^2)*P(n-1,x) - (n*x+(8*n-12)*x^2+(4*n-12)*x^3)*d/dx P(n-1,x) + (x^2+8*x^3+4*x^4)*d^2/dx^2 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 + (3*n^2 - n)*x - (4*n - 4)*x^2)*P[n - 1, x] - (n*x + (8*n - 12)*x^2 + (4*n - 12)*x^3)*D[P[n - 1, x], x] + (x^2 + 8*x^3 + 4*x^4)*D[P[n - 1, x], {x, 2}])/(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 + (3*n^2 - n)*x - (4*n - 4)*x^2)*v[i - 1] - (n*x + (8*n - 12)*x^2 + (4*n - 12)*x^3)*deriv(v[i - 1], x) + (x^2 + 8*x^3 + 4*x^4)*deriv(deriv(v[i - 1], x), x))/(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:   2,    1;
  2:   4,    5,    1;
  3:   8,   16,    8,     1;
  4:  16,   44,   37,    11,     1;
  5:  32,  112,  134,    67,    14,    1;
  6:  64,  272,  424,   305,   106,   17,    1;
  7: 128,  640, 1232,  1168,   584,  154,   20,   1;
  8: 256, 1472, 3376,  3992,  2641,  998,  211,  23,  1;
  9: 512, 3328, 8864, 12592, 10442, 5221, 1574, 277, 26, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (n+(3*n^2-n)*x-(4*n-4)*x^2)*P(n-1,x) - (n*x+(8*n-12)*x^2+(4*n-12)*x^3)*d/dx P(n-1,x) + (x^2+8*x^3+4*x^4)*d^2/dx^2 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 + (3*n^2 - n)*x - (4*n - 4)*x^2)*P[n - 1, x] - (n*x + (8*n - 12)*x^2 + (4*n - 12)*x^3)*D[P[n - 1, x], x] + (x^2 + 8*x^3 + 4*x^4)*D[P[n - 1, x], {x, 2}])/(n)]];
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] = ((n + (3*n^2 - n)*x - (4*n - 4)*x^2)*v[i - 1] - (n*x + (8*n - 12)*x^2 + (4*n - 12)*x^3)*deriv(v[i - 1], x) + (x^2 + 8*x^3 + 4*x^4)*deriv(deriv(v[i - 1], x), x))/(n)); 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)

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-635d5bcbc56b7dbf
• 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-635d5bcbc56b7dbf
• 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-635d5bcbc56b7dbf

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

• preserver_image reversal, shift 0 e-635d5bcbc56b7dbf

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-81aa6611c032ec70 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

Same fingerprint (1)

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

• A121574: Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).

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