A039756 → (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,  36,   117,     16;
  4: 1,  64,   522,    640,      25;
  5: 1, 100,  1530,   5280,    3025,       36;
  6: 1, 144,  3555,  24320,   44275,    13104,       49;
  7: 1, 196,  7119,  81200,  310275,   331884,    53557,       64;
  8: 1, 256, 12852, 220416, 1452150,  3459456,  2312212,   209920,     81;
  9: 1, 324, 21492, 517440, 5240550, 22279320, 35273140, 15310080, 797121, 100;

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = -(4*n^2-1+(32*n^3-48*n^2+24*n-36)*x-(144*n^4-320*n^3+152*n^2+16*n-15)*x^2)*P(n-2,x) - (4*x-(32*n^3-16*n^2-8*n+44)*x^2-(64*n^4-352*n^3+224*n^2+8*n-20)*x^3)*d/dx P(n-2,x) + (8*x^3-(64*n^3-48*n^2+4)*x^4)*d^2/dx^2 P(n-2,x) + (8*n^2-4*n+(32*n^3-48*n^2+24*n-4)*x)*P(n-1,x) + (4*x-(32*n^3-16*n^2-8*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 1 + (32*n^3 - 48*n^2 + 24*n - 36)*x - (144*n^4 - 320*n^3 + 152*n^2 + 16*n - 15)*x^2)*P[n - 2, x] - (4*x - (32*n^3 - 16*n^2 - 8*n + 44)*x^2 - (64*n^4 - 352*n^3 + 224*n^2 + 8*n - 20)*x^3)*D[P[n - 2, x], x] + (8*x^3 - (64*n^3 - 48*n^2 + 4)*x^4)*D[P[n - 2, x], {x, 2}] + (8*n^2 - 4*n + (32*n^3 - 48*n^2 + 24*n - 4)*x)*P[n - 1, x] + (4*x - (32*n^3 - 16*n^2 - 8*n + 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 4*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2 - 1 + (32*n^3 - 48*n^2 + 24*n - 36)*x - (144*n^4 - 320*n^3 + 152*n^2 + 16*n - 15)*x^2)*v[i - 2] - (4*x - (32*n^3 - 16*n^2 - 8*n + 44)*x^2 - (64*n^4 - 352*n^3 + 224*n^2 + 8*n - 20)*x^3)*deriv(v[i - 2], x) + (8*x^3 - (64*n^3 - 48*n^2 + 4)*x^4)*deriv(deriv(v[i - 2], x), x) + (8*n^2 - 4*n + (32*n^3 - 48*n^2 + 24*n - 4)*x)*v[i - 1] + (4*x - (32*n^3 - 16*n^2 - 8*n + 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 4*n + 1)); 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-efa089dc68f7f884
• 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-efa089dc68f7f884
• 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-efa089dc68f7f884
• 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-efa089dc68f7f884

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-efa089dc68f7f884

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

  • premise (sequence): Triangle of B-analogs of Stirling numbers of 2nd kind.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-7519e7535554cb98 root: theorem check

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

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.