A035607 → x ↦ x+1

OEIS matches ?

• A007483: row sums, 14 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:    1;
  2:    3,    2;
  3:    7,    8,     2;
  4:   17,   28,    14,    2;
  5:   41,   88,    66,   20,    2;
  6:   99,  262,   262,  122,   26,    2;
  7:  239,  752,   940,  592,  196,   32,   2;
  8:  577, 2104,  3156, 2508, 1132,  288,  38,  2;
  9: 1393, 5776, 10108, 9704, 5560, 1936, 398, 44, 2;

The row polynomials satisfy (n-2)*P(n,x) = (2*n^2-5*n+2)*P(n-1,x) - (8*n-20+(8*n-23)*x+(n-4)*x^2)*d/dx P(n-1,x) + (8+16*x+9*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 3 + 2*x.

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2)*P[n - 1, x] - (8*n - 20 + (8*n - 23)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (8 + 16*x + 9*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 3 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2)*v[i - 1] - (8*n - 20 + (8*n - 23)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (8 + 16*x + 9*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 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)

Proved properties

• 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-5cc1f9826e25e5f1
• 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-5cc1f9826e25e5f1
• 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-5cc1f9826e25e5f1
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-15636530e2574f08, e-5cc1f9826e25e5f1
• Real-rooted. Every row P_n, n >= 1, 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-15636530e2574f08
• Coefficients of one sign. Every nonzero row, n >= 1: 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-15636530e2574f08
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-15636530e2574f08

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

• preserver_image affine, shift 0 e-15636530e2574f08

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-f5199491b1877911 root: theorem check

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

• preserver_image affine, shift 0 e-5cc1f9826e25e5f1

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…
    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-1e993b9b3c7faf34 root: theorem check

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

Related sequences ?

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

• A035607 → basis A007318 (15 rows compared; no all-index equality certificate was found)

Same fingerprint (2)

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

• A122542 → x ↦ x/(x+1)
• A113413 → 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