A081582: Pascal-(1,7,1) array.

OEIS matches ?

• A081582: reversed triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A015519: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A081582: triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

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,  1;
  2: 1,  9,    1;
  3: 1, 17,   17,     1;
  4: 1, 25,   97,    25,     1;
  5: 1, 33,  241,   241,    33,     1;
  6: 1, 41,  449,  1161,   449,    41,     1;
  7: 1, 49,  721,  3297,  3297,   721,    49,    1;
  8: 1, 57, 1057,  7161, 14721,  7161,  1057,   57,  1;
  9: 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 7*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 7*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 7*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: 1,  1;
  2: 1,  9,    1;
  3: 1, 17,   17,     1;
  4: 1, 25,   97,    25,     1;
  5: 1, 33,  241,   241,    33,     1;
  6: 1, 41,  449,  1161,   449,    41,     1;
  7: 1, 49,  721,  3297,  3297,   721,    49,    1;
  8: 1, 57, 1057,  7161, 14721,  7161,  1057,   57,  1;
  9: 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 7*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 7*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 7*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,  1;
  2: 1,  9,    1;
  3: 1, 17,   17,     1;
  4: 1, 25,   97,    25,     1;
  5: 1, 33,  241,   241,    33,     1;
  6: 1, 41,  449,  1161,   449,    41,     1;
  7: 1, 49,  721,  3297,  3297,   721,    49,    1;
  8: 1, 57, 1057,  7161, 14721,  7161,  1057,   57,  1;
  9: 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 7*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 7*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 7*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-a31f5c33917b15a7
• 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-7d90ede2da8763e3, e-a31f5c33917b15a7
• 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-7d90ede2da8763e3, e-a31f5c33917b15a7
• 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-a31f5c33917b15a7
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-7 - sqrt(56), -7 + sqrt(56)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] 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-216cbdf61e3ed56f
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-7 - sqrt(56), -7 + sqrt(56)] to roots in [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]: every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has an image with all roots in I = [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]
  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-216cbdf61e3ed56f
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-7 - sqrt(56), 0] to roots in [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]: every PF input with all roots in [-7 - sqrt(56), 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-216cbdf61e3ed56f
• 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-7d90ede2da8763e3

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-216cbdf61e3ed56f root: theorem check

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

• definition_equality e-8666650c0d0b3b54 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image reversal, shift 0 e-7d90ede2da8763e3

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Pascal-(1,7,1) array.
    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-a31f5c33917b15a7 root: theorem check

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

• ↵ theorem_check liu_wang_derivative_lag e-a31f5c33917b15a7 (expanded above)

Used in 14 proof steps

• e-62ffc182dddb0e7d basis_map (A008277) → basis_map_A008277 of Pascal-(1,7,1) array.
• e-85efc751f4220bb8 basis_map (A013611) → basis_map_A013611 of Pascal-(1,7,1) array.
• e-be875bddda915b22 basis_map (A013610) → basis_map_A013610 of Pascal-(1,7,1) array.
• e-c5fd3da4c5a13f73 basis_map (A007318) → basis_map_A007318 of Pascal-(1,7,1) array.
• e-fa71667350448e54 basis_map (A013609) → basis_map_A013609 of Pascal-(1,7,1) array.
• e-c807fb938e5557d1 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Pascal-(1,7,1) array.
• e-03aefd7a03322503 preserver_image (affine, shift 0) → basis_map_A007318 of Pascal-(1,7,1) array.
• e-73cab1ceabab0270 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Pascal-(1,7,1) array.
• e-7d90ede2da8763e3 preserver_image (reversal, shift 0) → Pascal-(1,7,1) array.
• e-cc281e0cede67e51 preserver_image (derivative_polynomial, shift 0) → derivative of Pascal-(1,7,1) array.
• e-e37cbf39b02138a5 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Pascal-(1,7,1) array.
• e-ee1a409b71674b97 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Pascal-(1,7,1) array.
• e-1c0801c63e403da5 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Pascal-(1,7,1) array.
• e-bace45a12b7c6e7a veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Pascal-(1,7,1) array.

Related sequences ?

No candidate equalities or related sequences.

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