A081579 → (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,  24,    9;
  3: 1,  44,   99,    16;
  4: 1,  64,  414,   256,     25;
  5: 1,  84,  954,  1696,    525,     36;
  6: 1, 104, 1719,  6336,   4775,    936,     49;
  7: 1, 124, 2709, 16176,  25275,  10836,   1519,    64;
  8: 1, 144, 3924, 33216,  90150,  74736,  21364,  2304,   81;
  9: 1, 164, 5364, 59456, 243150, 350136, 182084, 38144, 3321, 100;

The row polynomials satisfy -(n^2+n+(15*n^3-2*n^2-13*n)*x+(25*n^4-50*n^2-7*n)*x^2-(25*n^4+120*n^3+158*n^2+81*n)*x^3+(5*n^3+9*n^2)*x^4)*P(n,x) = -(n^2+n+(15*n^3+38*n^2-9*n-36)*x+(25*n^4+250*n^3+140*n^2-357*n-90)*x^2+(225*n^4+755*n^3+227*n^2-945*n-774)*x^3-(120*n^3+466*n^2+618*n+318)*x^4+(10*n^2+28*n+18)*x^5)*P(n-1,x) + ((n+1)*x+(15*n^2+47*n+27)*x^2+(25*n^3+385*n^2+687*n+242)*x^3+(450*n^3+2080*n^2+2967*n+1422)*x^4+(25*n^3+25*n^2-88*n-83)*x^5-(5*n^2+14*n+9)*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 + (15*n^3 + 38*n^2 - 9*n - 36)*x + (25*n^4 + 250*n^3 + 140*n^2 - 357*n - 90)*x^2 + (225*n^4 + 755*n^3 + 227*n^2 - 945*n - 774)*x^3 - (120*n^3 + 466*n^2 + 618*n + 318)*x^4 + (10*n^2 + 28*n + 18)*x^5)*P[n - 1, x] + ((n + 1)*x + (15*n^2 + 47*n + 27)*x^2 + (25*n^3 + 385*n^2 + 687*n + 242)*x^3 + (450*n^3 + 2080*n^2 + 2967*n + 1422)*x^4 + (25*n^3 + 25*n^2 - 88*n - 83)*x^5 - (5*n^2 + 14*n + 9)*x^6)*D[P[n - 1, x], x])/(-n^2 - n - (15*n^3 - 2*n^2 - 13*n)*x - (25*n^4 - 50*n^2 - 7*n)*x^2 + (25*n^4 + 120*n^3 + 158*n^2 + 81*n)*x^3 - (5*n^3 + 9*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 + (15*n^3 + 38*n^2 - 9*n - 36)*x + (25*n^4 + 250*n^3 + 140*n^2 - 357*n - 90)*x^2 + (225*n^4 + 755*n^3 + 227*n^2 - 945*n - 774)*x^3 - (120*n^3 + 466*n^2 + 618*n + 318)*x^4 + (10*n^2 + 28*n + 18)*x^5)*v[i - 1] + ((n + 1)*x + (15*n^2 + 47*n + 27)*x^2 + (25*n^3 + 385*n^2 + 687*n + 242)*x^3 + (450*n^3 + 2080*n^2 + 2967*n + 1422)*x^4 + (25*n^3 + 25*n^2 - 88*n - 83)*x^5 - (5*n^2 + 14*n + 9)*x^6)*deriv(v[i - 1], x))/(-n^2 - n - (15*n^3 - 2*n^2 - 13*n)*x - (25*n^4 - 50*n^2 - 7*n)*x^2 + (25*n^4 + 120*n^3 + 158*n^2 + 81*n)*x^3 - (5*n^3 + 9*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-83f3bc33323b8934
• 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-83f3bc33323b8934
• 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-83f3bc33323b8934
• 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-83f3bc33323b8934

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-83f3bc33323b8934

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

  • premise (sequence): Pascal-(1,4,1) 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-01acb0cb09e95244 root: theorem check

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

    • preserver_image reversal, shift 0 e-93c62bb8c1382161

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,4,1) 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-01acb0cb09e95244 (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.