A113413: A Riordan array of coordination sequences.

OEIS matches ?

• A001333: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A113413: triangle, 69 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 ?
• A035607: reversed triangle, 71 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 ?
• A078057: 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 ?

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: 2,  1;
  2: 2,  4,   1;
  3: 2,  8,   6,   1;
  4: 2, 12,  18,   8,    1;
  5: 2, 16,  38,  32,   10,   1;
  6: 2, 20,  66,  88,   50,  12,   1;
  7: 2, 24, 102, 192,  170,  72,  14,   1;
  8: 2, 28, 146, 360,  450, 292,  98,  16,  1;
  9: 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

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

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 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: 1,  2;
  2: 1,  4,   2;
  3: 1,  6,   8,   2;
  4: 1,  8,  18,  12,   2;
  5: 1, 10,  32,  38,  16,    2;
  6: 1, 12,  50,  88,  66,   20,   2;
  7: 1, 14,  72, 170, 192,  102,  24,   2;
  8: 1, 16,  98, 292, 450,  360, 146,  28,  2;
  9: 1, 18, 128, 462, 912, 1002, 608, 198, 32, 2;

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

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 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)

Triangle begins:
  0: 1;
  1: 2,  1;
  2: 2,  4,   1;
  3: 2,  8,   6,   1;
  4: 2, 12,  18,   8,    1;
  5: 2, 16,  38,  32,   10,   1;
  6: 2, 20,  66,  88,   50,  12,   1;
  7: 2, 24, 102, 192,  170,  72,  14,   1;
  8: 2, 28, 146, 360,  450, 292,  98,  16,  1;
  9: 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

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

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 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)

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-7fe1799d89540b32
• 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-7fe1799d89540b32
• 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-7fe1799d89540b32
• 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-7fe1799d89540b32
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(2), -1 + sqrt(2)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] 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-dbb2ac730432d636
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(2), -1 + sqrt(2)] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]
  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-dbb2ac730432d636
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(2), 0] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every PF input with all roots in [-1 - sqrt(2), 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-dbb2ac730432d636

Proof chains

2 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-dbb2ac730432d636 root: theorem check

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

• theorem_check liu_wang_derivative_lag e-7fe1799d89540b32 root: theorem check

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

Used in 14 proof steps

• e-14cc72d2ba3c3aa2 basis_map (A013611) → basis_map_A013611 of A Riordan array of coordination sequences.
• e-3d8a64e4b99d997d basis_map (A013610) → basis_map_A013610 of A Riordan array of coordination sequences.
• e-69ce91caff12536e basis_map (A007318) → basis_map_A007318 of A Riordan array of coordination sequences.
• e-7383986e8ecff3a4 basis_map (A013609) → basis_map_A013609 of A Riordan array of coordination sequences.
• e-b8068c9d913440aa basis_map (A008277) → basis_map_A008277 of A Riordan array of coordination sequences.
• e-73ab557b7db9dc16 preserver_image (affine, shift 0) → basis_map_A007318 of A Riordan array of coordination sequences.
• e-73b4fdfae3fbc1a0 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of A Riordan array of coordination sequences.
• e-79715410c2bf252f preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of A Riordan array of coordination sequences.
• e-cbf712b709a8ecc6 preserver_image (derivative_polynomial, shift 0) → derivative of A Riordan array of coordination sequences.
• e-de1b791dfb436520 preserver_image (reversal, shift 0) → reversal of A Riordan array of coordination sequences.
• e-ea263f6af5fbc945 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of A Riordan array of coordination sequences.
• e-ef030cea3fd1d83e preserver_image (euler_polynomial, shift 0) → theta_plus_one of A Riordan array of coordination sequences.
• e-1e52280d1d1f6458 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of A Riordan array of coordination sequences.
• e-af7260029131ec6a veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of A Riordan array of coordination sequences.

Related sequences ?

Same fingerprint (6)

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

• A122542: Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0…
• A035607: Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
• A113413 → reverse
• A122542 → reverse
• A035607 → reverse
• A266213: Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, …

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