A081582 → 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:  2,    1;
  2: 10,   11,     1;
  3: 18,   51,    20,      1;
  4: 26,  219,   172,     29,      1;
  5: 34,  515,   964,    373,     38,      1;
  6: 42,  939,  3932,   2957,    654,     47,     1;
  7: 50, 1491, 10612,  16485,   6902,   1015,    56,    1;
  8: 58, 2171, 22540,  66045,  50526,  13503,  1456,   65,  1;
  9: 66, 2979, 41252, 193109, 269766, 124551, 23464, 1977, 74, 1;

The row polynomials satisfy -(8*n^3-22*n^2+14*n-(8*n^3-35*n^2+72*n)*x-(28*n^2-69*n)*x^2-(n^2+26*n)*x^3-n*x^4)*P(n,x) = -(8*n^3-14*n^2+6*n+(120*n^3-429*n^2+219*n)*x+(412*n^2-1034*n+659)*x^2+(15*n^2+363*n-391)*x^3+(14*n-43)*x^4-x^5)*P(n-1,x) + ((8*n^2-6*n)*x+(240*n^2-151*n-14)*x^2+(8*n^2+865*n-391)*x^3+(59*n+857)*x^4+(n+59)*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[(-(8*n^3 - 14*n^2 + 6*n + (120*n^3 - 429*n^2 + 219*n)*x + (412*n^2 - 1034*n + 659)*x^2 + (15*n^2 + 363*n - 391)*x^3 + (14*n - 43)*x^4 - x^5)*P[n - 1, x] + ((8*n^2 - 6*n)*x + (240*n^2 - 151*n - 14)*x^2 + (8*n^2 + 865*n - 391)*x^3 + (59*n + 857)*x^4 + (n + 59)*x^5 + x^6)*D[P[n - 1, x], x])/(-8*n^3 + 22*n^2 - 14*n + (8*n^3 - 35*n^2 + 72*n)*x + (28*n^2 - 69*n)*x^2 + (n^2 + 26*n)*x^3 + n*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] = (-(8*n^3 - 14*n^2 + 6*n + (120*n^3 - 429*n^2 + 219*n)*x + (412*n^2 - 1034*n + 659)*x^2 + (15*n^2 + 363*n - 391)*x^3 + (14*n - 43)*x^4 - x^5)*v[i - 1] + ((8*n^2 - 6*n)*x + (240*n^2 - 151*n - 14)*x^2 + (8*n^2 + 865*n - 391)*x^3 + (59*n + 857)*x^4 + (n + 59)*x^5 + x^6)*deriv(v[i - 1], x))/(-8*n^3 + 22*n^2 - 14*n + (8*n^3 - 35*n^2 + 72*n)*x + (28*n^2 - 69*n)*x^2 + (n^2 + 26*n)*x^3 + n*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-e37cbf39b02138a5
• 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-e37cbf39b02138a5
• 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-e37cbf39b02138a5
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-e37cbf39b02138a5

Proof chains

1 incoming proof step. 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-e37cbf39b02138a5

  Laguerre’s theorem on differential operators [SP14] 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 ?

    • 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 (expanded above)

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.