A008292 → D+1

OEIS matches ?

• A357711: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A188588: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A001710: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:
   1:    1;
   2:    2,     1;
   3:    5,     6,       1;
   4:   12,    33,      14,       1;
   5:   27,   158,     144,      30,       1;
   6:   58,   661,    1208,     530,      62,       1;
   7:  121,  2502,    8439,    7180,    1791,     126,      1;
   8:  248,  8833,   51150,   78095,   37084,    5775,    254,     1;
   9:  503, 29718,  279310,  712994,  597360,  175882,  18122,   510,    1;
  10: 1014, 96693, 1413416, 5696608, 7862124, 4041506, 790072, 55944, 1022, 1;

The row polynomials satisfy (n-3+2*x)*P(n,x) = -(2*n-4+(2*n^2-11*n+18)*x+(n^3-8*n^2+25*n-32)*x^2+(3*n^2-16*n+22)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((n-2)*x+(n^2-6*n+10)*x^2-(n^2-8*n+16)*x^3-(3*n-10)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-8+(2*n^2-11*n+21)*x+(5*n-14)*x^2+2*x^3)*P(n-1,x) + ((n-3)*x-(n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (2*n^2 - 11*n + 18)*x + (n^3 - 8*n^2 + 25*n - 32)*x^2 + (3*n^2 - 16*n + 22)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((n - 2)*x + (n^2 - 6*n + 10)*x^2 - (n^2 - 8*n + 16)*x^3 - (3*n - 10)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 8 + (2*n^2 - 11*n + 21)*x + (5*n - 14)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 3)*x - (n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 3 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(2*n - 4 + (2*n^2 - 11*n + 18)*x + (n^3 - 8*n^2 + 25*n - 32)*x^2 + (3*n^2 - 16*n + 22)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((n - 2)*x + (n^2 - 6*n + 10)*x^2 - (n^2 - 8*n + 16)*x^3 - (3*n - 10)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 8 + (2*n^2 - 11*n + 21)*x + (5*n - 14)*x^2 + 2*x^3)*v[i - 1] + ((n - 3)*x - (n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 3 + 2*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• 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-667ea58aa192e8e7
• 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-667ea58aa192e8e7
• 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-667ea58aa192e8e7
• 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-667ea58aa192e8e7

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-667ea58aa192e8e7

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): eulerian
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ab3cdcfae289b48d root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers root_domain e-6c46c948823f0a8c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-c61ecf4ef1c45372

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): eulerian
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-ab3cdcfae289b48d (expanded above)
        • ↵ parent_provers root_domain e-6c46c948823f0a8c (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A173018 → D+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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.