A035607 → D+1

OEIS matches ?

No OEIS matches.

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:  5,   8,   2;
  4:  7,  22,  14,    2;
  5:  9,  44,  54,   20,    2;
  6: 11,  74, 146,  102,   26,    2;
  7: 13, 112, 314,  352,  166,   32,   2;
  8: 15, 158, 582,  938,  702,  246,  38,  2;
  9: 17, 212, 974, 2092, 2250, 1236, 342, 44, 2;

The row polynomials satisfy -(2*n^3-5*n^2+3*n-(2*n^3-11*n^2+16*n)*x-(5*n^2-22*n+16)*x^2-(n^2-6)*x^3-(n-2)*x^4)*P(n,x) = -(2*n^3-3*n^2+n+(6*n^3-15*n^2+2*n)*x+(15*n^2-30*n+16)*x^2+(3*n^2+8*n-6)*x^3+(3*n-2)*x^4)*P(n-1,x) + ((2*n^2-n)*x+(12*n^2-2)*x^2+(2*n^2+36*n-7)*x^3+(12*n+29)*x^4+(n+11)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 3*n^2 + n + (6*n^3 - 15*n^2 + 2*n)*x + (15*n^2 - 30*n + 16)*x^2 + (3*n^2 + 8*n - 6)*x^3 + (3*n - 2)*x^4)*P[n - 1, x] + ((2*n^2 - n)*x + (12*n^2 - 2)*x^2 + (2*n^2 + 36*n - 7)*x^3 + (12*n + 29)*x^4 + (n + 11)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^3 + 5*n^2 - 3*n + (2*n^3 - 11*n^2 + 16*n)*x + (5*n^2 - 22*n + 16)*x^2 + (n^2 - 6)*x^3 + (n - 2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 - 3*n^2 + n + (6*n^3 - 15*n^2 + 2*n)*x + (15*n^2 - 30*n + 16)*x^2 + (3*n^2 + 8*n - 6)*x^3 + (3*n - 2)*x^4)*v[i - 1] + ((2*n^2 - n)*x + (12*n^2 - 2)*x^2 + (2*n^2 + 36*n - 7)*x^3 + (12*n + 29)*x^4 + (n + 11)*x^5 + x^6)*deriv(v[i - 1], x))/(-2*n^3 + 5*n^2 - 3*n + (2*n^3 - 11*n^2 + 16*n)*x + (5*n^2 - 22*n + 16)*x^2 + (n^2 - 6)*x^3 + (n - 2)*x^4)); 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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-b2a6ce2f982337a1
• 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-b2a6ce2f982337a1
• 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-b2a6ce2f982337a1
• 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-01157a6073b8735e, e-b2a6ce2f982337a1
• Real-rooted. Every row P_n, n >= 1, 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-01157a6073b8735e
• 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-01157a6073b8735e
• 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-01157a6073b8735e

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 derivative_polynomial, shift 0 e-01157a6073b8735e

  Laguerre’s theorem on differential operators [SP14] 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 derivative_polynomial, shift 0 e-b2a6ce2f982337a1

  Laguerre’s theorem on differential operators [SP14] 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 ?

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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.