A227450 → reverse

OEIS matches ?

• A227450: 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 = 1 ?
• A001861: 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:    2;
  1:    4,     2;
  2:    8,    12,      2;
  3:   16,    48,     28,      2;
  4:   32,   160,    200,     60,       2;
  5:   64,   480,   1040,    720,     124,       2;
  6:  128,  1344,   4480,   5600,    2408,     252,      2;
  7:  256,  3584,  17024,  33600,   27216,    7728,    508,     2;
  8:  512,  9216,  59136, 169344,  222432,  124320,  24200,  1020,    2;
  9: 1024, 23040, 192000, 752640, 1460928, 1360800, 545680, 74640, 2044, 2;

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

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(2 + n*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = (2 + n*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); 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:
   1: 2;
   2: 2,    4;
   3: 2,   12,     8;
   4: 2,   28,    48,     16;
   5: 2,   60,   200,    160,      32;
   6: 2,  124,   720,   1040,     480,      64;
   7: 2,  252,  2408,   5600,    4480,    1344,    128;
   8: 2,  508,  7728,  27216,   33600,   17024,   3584,    256;
   9: 2, 1020, 24200, 124320,  222432,  169344,  59136,   9216,   512;
  10: 2, 2044, 74640, 545680, 1360800, 1460928, 752640, 192000, 23040, 1024;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(2 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = (2 + (n - 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-88af8bd6d056b310
• 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-88af8bd6d056b310
• 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-88af8bd6d056b310

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-88af8bd6d056b310

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-74361f30d2bc482d root: theorem check

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

Related sequences ?

Same fingerprint (1)

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

• A227450: Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.

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