A081578 → (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,  20,    9;
  3: 1,  36,   81,    16;
  4: 1,  52,  297,   208,     25;
  5: 1,  68,  657,  1168,    425,     36;
  6: 1,  84, 1161,  3920,   3225,    756,     49;
  7: 1, 100, 1809,  9488,  14825,   7236,   1225,    64;
  8: 1, 116, 2601, 18896,  48025,  42516,  14161,  1856,   81;
  9: 1, 132, 3537, 33168, 122025, 175716, 101577, 25152, 2673, 100;

The row polynomials satisfy -(n^2+n+(12*n^3-2*n^2-11*n)*x+(16*n^4-4*n^3-40*n^2-5*n)*x^2-(16*n^4+76*n^3+94*n^2+49*n)*x^3+(4*n^3+7*n^2)*x^4)*P(n,x) = -(n^2+n+(12*n^3+30*n^2-7*n-28)*x+(16*n^4+156*n^3+80*n^2-231*n-58)*x^2+(112*n^4+356*n^3+42*n^2-499*n-378)*x^3-(76*n^3+289*n^2+374*n+190)*x^4+(8*n^2+22*n+14)*x^5)*P(n-1,x) + ((n+1)*x+(12*n^2+37*n+21)*x^2+(16*n^3+240*n^2+422*n+146)*x^3+(224*n^3+1016*n^2+1398*n+658)*x^4+(16*n^3+16*n^2-55*n-51)*x^5-(4*n^2+11*n+7)*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 + (12*n^3 + 30*n^2 - 7*n - 28)*x + (16*n^4 + 156*n^3 + 80*n^2 - 231*n - 58)*x^2 + (112*n^4 + 356*n^3 + 42*n^2 - 499*n - 378)*x^3 - (76*n^3 + 289*n^2 + 374*n + 190)*x^4 + (8*n^2 + 22*n + 14)*x^5)*P[n - 1, x] + ((n + 1)*x + (12*n^2 + 37*n + 21)*x^2 + (16*n^3 + 240*n^2 + 422*n + 146)*x^3 + (224*n^3 + 1016*n^2 + 1398*n + 658)*x^4 + (16*n^3 + 16*n^2 - 55*n - 51)*x^5 - (4*n^2 + 11*n + 7)*x^6)*D[P[n - 1, x], x])/(-n^2 - n - (12*n^3 - 2*n^2 - 11*n)*x - (16*n^4 - 4*n^3 - 40*n^2 - 5*n)*x^2 + (16*n^4 + 76*n^3 + 94*n^2 + 49*n)*x^3 - (4*n^3 + 7*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 + (12*n^3 + 30*n^2 - 7*n - 28)*x + (16*n^4 + 156*n^3 + 80*n^2 - 231*n - 58)*x^2 + (112*n^4 + 356*n^3 + 42*n^2 - 499*n - 378)*x^3 - (76*n^3 + 289*n^2 + 374*n + 190)*x^4 + (8*n^2 + 22*n + 14)*x^5)*v[i - 1] + ((n + 1)*x + (12*n^2 + 37*n + 21)*x^2 + (16*n^3 + 240*n^2 + 422*n + 146)*x^3 + (224*n^3 + 1016*n^2 + 1398*n + 658)*x^4 + (16*n^3 + 16*n^2 - 55*n - 51)*x^5 - (4*n^2 + 11*n + 7)*x^6)*deriv(v[i - 1], x))/(-n^2 - n - (12*n^3 - 2*n^2 - 11*n)*x - (16*n^4 - 4*n^3 - 40*n^2 - 5*n)*x^2 + (16*n^4 + 76*n^3 + 94*n^2 + 49*n)*x^3 - (4*n^3 + 7*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-80bca760e085f131
• 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-80bca760e085f131
• 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-80bca760e085f131
• 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-80bca760e085f131

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-80bca760e085f131

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

  • premise (sequence): Pascal-(1,3,1) 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-fba968684828f914 root: theorem check

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

    • preserver_image reversal, shift 0 e-86685874cd5f1179

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,3,1) 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-fba968684828f914 (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.