A144438 → (θ+1)²

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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,   56,    126,       16;
  4: 1,  132,    801,      528,       25;
  5: 1,  288,   3717,     6608,     1800,       36;
  6: 1,  604,  14688,    54288,    40800,     5436,       49;
  7: 1, 1240,  52866,   354320,   553625,   211464,    15190,      64;
  8: 1, 2516, 179487,  2007296,  5621075,  4516416,   977207,   40256,     81;
  9: 1, 5072, 586539, 10356784, 47477400, 68367456, 31717651, 4170944, 102708, 100;

The row polynomials satisfy P(n,x) = 11*x^2*P(n-4,x) + 5*x^3*d/dx P(n-4,x) - (4*x-(20*n-15)*x^2)*P(n-3,x) + (5*x^2+(9*n-23)*x^3)*d/dx P(n-3,x) + (5*x^3-5*x^4)*d^2/dx^2 P(n-3,x) - ((4*n-8)*x-(9*n^2-2*n-7)*x^2)*P(n-2,x) - (2*x-(4*n+4)*x^2-(4*n^2-14*n-13)*x^3)*d/dx P(n-2,x) - (x^2-(4*n+4)*x^3+(4*n+3)*x^4)*d^2/dx^2 P(n-2,x) + (1+4*n*x)*P(n-1,x) + (2*x-(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 20*x + 9*x^2, P(3,x) = 1 + 56*x + 126*x^2 + 16*x^3.

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 20*x + 9*x^2;
P[3, x_] := 1 + 56*x + 126*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[11*x^2*P[n - 4, x] + 5*x^3*D[P[n - 4, x], x] - (4*x - (20*n - 15)*x^2)*P[n - 3, x] + (5*x^2 + (9*n - 23)*x^3)*D[P[n - 3, x], x] + (5*x^3 - 5*x^4)*D[P[n - 3, x], {x, 2}] - ((4*n - 8)*x - (9*n^2 - 2*n - 7)*x^2)*P[n - 2, x] - (2*x - (4*n + 4)*x^2 - (4*n^2 - 14*n - 13)*x^3)*D[P[n - 2, x], x] - (x^2 - (4*n + 4)*x^3 + (4*n + 3)*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 4*n*x)*P[n - 1, x] + (2*x - (4*n + 4)*x^2)*D[P[n - 1, x], x]];
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; v[3] = 1 + 20*x + 9*x^2; v[4] = 1 + 56*x + 126*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = 11*x^2*v[i - 4] + 5*x^3*deriv(v[i - 4], x) - (4*x - (20*n - 15)*x^2)*v[i - 3] + (5*x^2 + (9*n - 23)*x^3)*deriv(v[i - 3], x) + (5*x^3 - 5*x^4)*deriv(deriv(v[i - 3], x), x) - ((4*n - 8)*x - (9*n^2 - 2*n - 7)*x^2)*v[i - 2] - (2*x - (4*n + 4)*x^2 - (4*n^2 - 14*n - 13)*x^3)*deriv(v[i - 2], x) - (x^2 - (4*n + 4)*x^3 + (4*n + 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 4*n*x)*v[i - 1] + (2*x - (4*n + 4)*x^2)*deriv(v[i - 1], x)); 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-55abd557d63ba53c
• 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-55abd557d63ba53c
• 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-55abd557d63ba53c
• 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-55abd557d63ba53c

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-55abd557d63ba53c

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

  • premise (sequence): Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
    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-d401dbe9093b268d root: theorem check

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

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.