A100862 → (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,  12,    9;
  3: 1,  24,   54,    16;
  4: 1,  40,  189,   160,     25;
  5: 1,  60,  495,   880,    375,     36;
  6: 1,  84, 1080,  3440,   3000,    756,     49;
  7: 1, 112, 2079, 10640,  16625,   8316,   1372,    64;
  8: 1, 144, 3654, 27776,  70875,  62496,  19894,  2304,   81;
  9: 1, 180, 5994, 63840, 247275, 356076, 195510, 42624, 3645, 100;

The row polynomials satisfy -(4+(6*n^2+10*n-20)*x+(n^4+6*n^3-n^2-54*n+28)*x^2-(n^4+14*n^3+43*n^2-30*n+16)*x^3+(4*n^3+30*n^2+2*n)*x^4-4*n^2*x^5)*P(n,x) = -(4+(6*n^2+26*n-20)*x+(n^4+16*n^3+33*n^2-78*n-8)*x^2+(n^5+9*n^4+11*n^3-85*n^2-48*n-12)*x^3-(5*n^4+40*n^3+57*n^2+30*n+8)*x^4+(8*n^3+42*n^2+62*n+28)*x^5-(4*n^2+8*n+4)*x^6)*P(n-1,x) + (8*x^2+(12*n^2+44*n)*x^3+(2*n^4+20*n^3+58*n^2-8)*x^4-(8*n^3+60*n^2+76*n+32)*x^5+(8*n^2+16*n+8)*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[(-(4 + (6*n^2 + 26*n - 20)*x + (n^4 + 16*n^3 + 33*n^2 - 78*n - 8)*x^2 + (n^5 + 9*n^4 + 11*n^3 - 85*n^2 - 48*n - 12)*x^3 - (5*n^4 + 40*n^3 + 57*n^2 + 30*n + 8)*x^4 + (8*n^3 + 42*n^2 + 62*n + 28)*x^5 - (4*n^2 + 8*n + 4)*x^6)*P[n - 1, x] + (8*x^2 + (12*n^2 + 44*n)*x^3 + (2*n^4 + 20*n^3 + 58*n^2 - 8)*x^4 - (8*n^3 + 60*n^2 + 76*n + 32)*x^5 + (8*n^2 + 16*n + 8)*x^6)*D[P[n - 1, x], x])/(-4 - (6*n^2 + 10*n - 20)*x - (n^4 + 6*n^3 - n^2 - 54*n + 28)*x^2 + (n^4 + 14*n^3 + 43*n^2 - 30*n + 16)*x^3 - (4*n^3 + 30*n^2 + 2*n)*x^4 + 4*n^2*x^5)]];
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] = (-(4 + (6*n^2 + 26*n - 20)*x + (n^4 + 16*n^3 + 33*n^2 - 78*n - 8)*x^2 + (n^5 + 9*n^4 + 11*n^3 - 85*n^2 - 48*n - 12)*x^3 - (5*n^4 + 40*n^3 + 57*n^2 + 30*n + 8)*x^4 + (8*n^3 + 42*n^2 + 62*n + 28)*x^5 - (4*n^2 + 8*n + 4)*x^6)*v[i - 1] + (8*x^2 + (12*n^2 + 44*n)*x^3 + (2*n^4 + 20*n^3 + 58*n^2 - 8)*x^4 - (8*n^3 + 60*n^2 + 76*n + 32)*x^5 + (8*n^2 + 16*n + 8)*x^6)*deriv(v[i - 1], x))/(-4 - (6*n^2 + 10*n - 20)*x - (n^4 + 6*n^3 - n^2 - 54*n + 28)*x^2 + (n^4 + 14*n^3 + 43*n^2 - 30*n + 16)*x^3 - (4*n^3 + 30*n^2 + 2*n)*x^4 + 4*n^2*x^5)); 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-46744df2075291bc
• 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-46744df2075291bc
• 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-46744df2075291bc
• 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-46744df2075291bc

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-46744df2075291bc

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
    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-e7942008ad3e1420 root: theorem check

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

    • preserver_image reversal, shift 0 e-7078061f6d32a158

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
        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-e7942008ad3e1420 (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.