A081577 → (θ+1)²

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,   4;
  2: 1,  16,    9;
  3: 1,  28,   63,    16;
  4: 1,  40,  198,   160,    25;
  5: 1,  52,  414,   736,   325,    36;
  6: 1,  64,  711,  2176,  1975,   576,    49;
  7: 1,  76, 1089,  4912,  7675,  4356,   931,    64;
  8: 1,  88, 1548,  9376, 22150, 21096,  8428,  1408,   81;
  9: 1, 100, 2088, 16000, 52150, 75096, 49000, 14848, 2025, 100;

The row polynomials satisfy -(n^2+n+(9*n^3-2*n^2-9*n)*x+(9*n^4-6*n^3-30*n^2-3*n)*x^2-(9*n^4+42*n^3+46*n^2+25*n)*x^3+(3*n^3+5*n^2)*x^4)*P(n,x) = -(n^2+n+(9*n^3+22*n^2-5*n-20)*x+(9*n^4+84*n^3+36*n^2-133*n-34)*x^2+(45*n^4+129*n^3-37*n^2-229*n-150)*x^3-(42*n^3+154*n^2+190*n+94)*x^4+(6*n^2+16*n+10)*x^5)*P(n-1,x) + ((n+1)*x+(9*n^2+27*n+15)*x^2+(9*n^3+129*n^2+221*n+74)*x^3+(90*n^3+396*n^2+509*n+230)*x^4+(9*n^3+9*n^2-30*n-27)*x^5-(3*n^2+8*n+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[(-(n^2 + n + (9*n^3 + 22*n^2 - 5*n - 20)*x + (9*n^4 + 84*n^3 + 36*n^2 - 133*n - 34)*x^2 + (45*n^4 + 129*n^3 - 37*n^2 - 229*n - 150)*x^3 - (42*n^3 + 154*n^2 + 190*n + 94)*x^4 + (6*n^2 + 16*n + 10)*x^5)*P[n - 1, x] + ((n + 1)*x + (9*n^2 + 27*n + 15)*x^2 + (9*n^3 + 129*n^2 + 221*n + 74)*x^3 + (90*n^3 + 396*n^2 + 509*n + 230)*x^4 + (9*n^3 + 9*n^2 - 30*n - 27)*x^5 - (3*n^2 + 8*n + 5)*x^6)*D[P[n - 1, x], x])/(-n^2 - n - (9*n^3 - 2*n^2 - 9*n)*x - (9*n^4 - 6*n^3 - 30*n^2 - 3*n)*x^2 + (9*n^4 + 42*n^3 + 46*n^2 + 25*n)*x^3 - (3*n^3 + 5*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] = (-(n^2 + n + (9*n^3 + 22*n^2 - 5*n - 20)*x + (9*n^4 + 84*n^3 + 36*n^2 - 133*n - 34)*x^2 + (45*n^4 + 129*n^3 - 37*n^2 - 229*n - 150)*x^3 - (42*n^3 + 154*n^2 + 190*n + 94)*x^4 + (6*n^2 + 16*n + 10)*x^5)*v[i - 1] + ((n + 1)*x + (9*n^2 + 27*n + 15)*x^2 + (9*n^3 + 129*n^2 + 221*n + 74)*x^3 + (90*n^3 + 396*n^2 + 509*n + 230)*x^4 + (9*n^3 + 9*n^2 - 30*n - 27)*x^5 - (3*n^2 + 8*n + 5)*x^6)*deriv(v[i - 1], x))/(-n^2 - n - (9*n^3 - 2*n^2 - 9*n)*x - (9*n^4 - 6*n^3 - 30*n^2 - 3*n)*x^2 + (9*n^4 + 42*n^3 + 46*n^2 + 25*n)*x^3 - (3*n^3 + 5*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-be4074d4da1f384a
• 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-be4074d4da1f384a
• 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-be4074d4da1f384a
• 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-be4074d4da1f384a

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 euler_polynomial, shift 0 e-be4074d4da1f384a

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
    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-1f8dc4b132ba6116 root: theorem check

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

    • preserver_image reversal, shift 0 e-8efc9aafb30c2198

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
        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-1f8dc4b132ba6116 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.