A113413 → basis A007318

OEIS matches ?

• A054458: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A055099: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A209696: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · 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:    7,     6,     1;
  3:   17,    23,     9,     1;
  4:   41,    76,    48,    12,     1;
  5:   99,   233,   204,    82,    15,    1;
  6:  239,   682,   765,   428,   125,   18,    1;
  7:  577,  1935,  2649,  1907,   775,  177,   21,   1;
  8: 1393,  5368,  8680,  7656,  4010, 1272,  238,  24,  1;
  9: 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + (2+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[(1 + x)*P[n - 2, x] + (2 + 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] = (1 + x)*v[i - 2] + (2 + 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:
  0:    1;
  1:    3,     1;
  2:    7,     6,     1;
  3:   17,    23,     9,     1;
  4:   41,    76,    48,    12,     1;
  5:   99,   233,   204,    82,    15,    1;
  6:  239,   682,   765,   428,   125,   18,    1;
  7:  577,  1935,  2649,  1907,   775,  177,   21,   1;
  8: 1393,  5368,  8680,  7656,  4010, 1272,  238,  24,  1;
  9: 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + (2+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[(1 + x)*P[n - 2, x] + (2 + x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (1 + x)*v[i - 2] + (2 + x)*v[i - 1]); 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)

Triangle begins:
   1: 1;
   2: 1,  3;
   3: 1,  6,   7;
   4: 1,  9,  23,   17;
   5: 1, 12,  48,   76,   41;
   6: 1, 15,  82,  204,  233,    99;
   7: 1, 18, 125,  428,  765,   682,   239;
   8: 1, 21, 177,  775, 1907,  2649,  1935,   577;
   9: 1, 24, 238, 1272, 4010,  7656,  8680,  5368,  1393;
  10: 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-2,x) + (2+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[(1 + x)*P[n - 2, x] + (2 + 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] = (1 + x)*v[i - 2] + (2 + 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-69ce91caff12536e
• 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-69ce91caff12536e, e-73ab557b7db9dc16
• 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-69ce91caff12536e, e-73ab557b7db9dc16
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-69ce91caff12536e, e-73ab557b7db9dc16
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-73ab557b7db9dc16

Proof chains

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

• basis_map A007318 e-69ce91caff12536e

  Geometric basis map (elementary) polytool ?

  • premise (sequence): A Riordan array of coordination sequences.
    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-7fe1799d89540b32 root: theorem check

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

• definition_equality e-1f6958db57a4f791 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-73ab557b7db9dc16

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): A Riordan array of coordination sequences.
    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-7fe1799d89540b32 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A035607 → x ↦ x/(x+1)

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