A249248 → reverse

OEIS matches ?

• A249248: reversed triangle, 58 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 ?

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: 3,   1;
  2: 3,   7,    1;
  3: 3,  22,   12,     1;
  4: 3,  40,   69,    18,     1;
  5: 3,  61,  241,   159,    25,     1;
  6: 3,  85,  582,   883,   310,    33,     1;
  7: 3, 112, 1155,  3393,  2465,   543,    42,    1;
  8: 3, 142, 2032,  9786, 13805,  5798,   882,   52,  1;
  9: 3, 175, 3294, 23368, 57521, 44253, 12110, 1354, 63, 1;

The row polynomials satisfy P(n,x) = (n+1)*x*P(n-2,x) + (1+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(n + 1)*x*P[n - 2, x] + (1 + 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; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1)*x*v[i - 2] + (1 + 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:
   1: 1;
   2: 1,  3;
   3: 1,  7,    3;
   4: 1, 12,   22,     3;
   5: 1, 18,   69,    40,     3;
   6: 1, 25,  159,   241,    61,     3;
   7: 1, 33,  310,   883,   582,    85,     3;
   8: 1, 42,  543,  2465,  3393,  1155,   112,    3;
   9: 1, 52,  882,  5798, 13805,  9786,  2032,  142,   3;
  10: 1, 63, 1354, 12110, 44253, 57521, 23368, 3294, 175, 3;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = n*x*P(n-2,x) + (1+x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.

P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[n*x*P[n - 2, x] + (1 + x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = n*x*v[i - 2] + (1 + x)*v[i - 1]); 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-aead51ce25e60b28
• 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-aead51ce25e60b28
• 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-aead51ce25e60b28

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-aead51ce25e60b28

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    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-2dcae0cd92b4b84a 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.

• A249248: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

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