A116412: Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

OEIS matches ?

• A116412: triangle, 57 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 ?
• A249251: reversed triangle, 57 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 ?
• A003688: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 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:   6,    6,     1;
  3:  12,   21,     9,     1;
  4:  24,   60,    45,    12,     1;
  5:  48,  156,   171,    78,    15,    1;
  6:  96,  384,   558,   372,   120,   18,    1;
  7: 192,  912,  1656,  1473,   690,  171,   21,   1;
  8: 384, 2112,  4608,  5160,  3225, 1152,  231,  24,  1;
  9: 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1;

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + x*P(n-2,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[(2 + x)*P[n - 1, x] + x*P[n - 2, 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] = (2 + x)*v[i - 1] + x*v[i - 2]); 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:   6,    6,     1;
  3:  12,   21,     9,     1;
  4:  24,   60,    45,    12,     1;
  5:  48,  156,   171,    78,    15,    1;
  6:  96,  384,   558,   372,   120,   18,    1;
  7: 192,  912,  1656,  1473,   690,  171,   21,   1;
  8: 384, 2112,  4608,  5160,  3225, 1152,  231,  24,  1;
  9: 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1;

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + x*P(n-2,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[(2 + x)*P[n - 1, x] + x*P[n - 2, 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] = (2 + x)*v[i - 1] + x*v[i - 2]); 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:
  0: 1;
  1: 1,  3;
  2: 1,  6,   6;
  3: 1,  9,  21,   12;
  4: 1, 12,  45,   60,   24;
  5: 1, 15,  78,  171,  156,    48;
  6: 1, 18, 120,  372,  558,   384,    96;
  7: 1, 21, 171,  690, 1473,  1656,   912,   192;
  8: 1, 24, 231, 1152, 3225,  5160,  4608,  2112,  384;
  9: 1, 27, 300, 1785, 6219, 13083, 16584, 12240, 4800, 768;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+x)*P(n-1,x) + x*P(n-2,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[(2 + x)*P[n - 1, x] + x*P[n - 2, x]];
Table[Reverse[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] = (2 + x)*v[i - 1] + x*v[i - 2]); 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-ee40b91a84fdd5b7
• 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-c36241f3b645d7b5, e-ee40b91a84fdd5b7
• 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-c36241f3b645d7b5, e-ee40b91a84fdd5b7
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-ee40b91a84fdd5b7
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(3), -1 + sqrt(3)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1 - sqrt(3), -1 + sqrt(3)] has a real-rooted image
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-1662d4b11e9f54c0
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(3), -1 + sqrt(3)] to roots in [-4 - 2 sqrt(3), -4 + 2 sqrt(3)]: every real polynomial with all roots in E = [-1 - sqrt(3), -1 + sqrt(3)] has an image with all roots in I = [-4 - 2 sqrt(3), -4 + 2 sqrt(3)]
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-1662d4b11e9f54c0
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(3), 0] to roots in [-4 - 2 sqrt(3), -4 + 2 sqrt(3)]: every PF input with all roots in [-1 - sqrt(3), 0] has a PF image; no larger [-R, 0] works
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-1662d4b11e9f54c0
• 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-c36241f3b645d7b5

Proof chains

4 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_theorem e-1662d4b11e9f54c0 root: theorem check

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

• definition_equality e-d34045cf0110047c root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image reversal, shift 0 e-c36241f3b645d7b5

  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-67b682ca6511d839 root: theorem check

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

    • preserver_image reversal, shift 0 e-c7b023a4533e2987

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
        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-ee40b91a84fdd5b7 root: theorem check

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

• ↵ theorem_check liu_wang_derivative_lag e-ee40b91a84fdd5b7 (expanded above)

Used in 14 proof steps

• e-0d5d22b8fd97251b basis_map (A013609) → basis_map_A013609 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-32d1efa81dadfdb2 basis_map (A007318) → basis_map_A007318 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-3771d6044a0ef815 basis_map (A013611) → basis_map_A013611 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-c215916a1d865227 basis_map (A008277) → basis_map_A008277 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-dddeff0317c778fa basis_map (A013610) → basis_map_A013610 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-0cc329e6d69bfe0a preserver_image (euler_polynomial, shift 0) → theta_plus_one of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-29ee48d67514cd97 preserver_image (derivative_polynomial, shift 0) → derivative of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-7fe4c3f24349ff5b preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-8755b73a3425b06c preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-a802d54c8a2137b8 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-c7b023a4533e2987 preserver_image (reversal, shift 0) → Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comme…
• e-c997a055dd6d21a0 preserver_image (affine, shift 0) → basis_map_A007318 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-593da3abd88a42f1 veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
• e-8a973afc86912433 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

Related sequences ?

Same fingerprint (1)

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

• A249251: 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