A152842 → θ+1

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,  6;
  2: 1,  8,   9;
  3: 1, 14,  45,   36;
  4: 1, 16,  66,   96,    45;
  5: 1, 22, 138,  360,   405,   162;
  6: 1, 24, 171,  544,   855,   648,   189;
  7: 1, 30, 279, 1228,  2895,  3726,  2457,   648;
  8: 1, 32, 324, 1600,  4430,  7200,  6804,  3456,   729;
  9: 1, 38, 468, 2896, 10430, 23148, 32004, 26784, 12393, 2430;

The row polynomials satisfy (4+(12*n+13)*x+(8*n^2+38*n)*x^2+(24*n^2+18*n-21)*x^3+(18*n^2-18*n)*x^4)*P(n,x) = (4+(12*n+41)*x+(8*n^2+102*n+158)*x^2+(56*n^2+318*n+280)*x^3+(138*n^2+444*n+222)*x^4+(144*n^2+270*n+63)*x^5+(54*n^2+54*n)*x^6)*P(n-2,x) + (4*x+22*x^2+36*x^3+18*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.

P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((4 + (12*n + 41)*x + (8*n^2 + 102*n + 158)*x^2 + (56*n^2 + 318*n + 280)*x^3 + (138*n^2 + 444*n + 222)*x^4 + (144*n^2 + 270*n + 63)*x^5 + (54*n^2 + 54*n)*x^6)*P[n - 2, x] + (4*x + 22*x^2 + 36*x^3 + 18*x^4)*P[n - 1, x])/(4 + (12*n + 13)*x + (8*n^2 + 38*n)*x^2 + (24*n^2 + 18*n - 21)*x^3 + (18*n^2 - 18*n)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((4 + (12*n + 41)*x + (8*n^2 + 102*n + 158)*x^2 + (56*n^2 + 318*n + 280)*x^3 + (138*n^2 + 444*n + 222)*x^4 + (144*n^2 + 270*n + 63)*x^5 + (54*n^2 + 54*n)*x^6)*v[i - 2] + (4*x + 22*x^2 + 36*x^3 + 18*x^4)*v[i - 1])/(4 + (12*n + 13)*x + (8*n^2 + 38*n)*x^2 + (24*n^2 + 18*n - 21)*x^3 + (18*n^2 - 18*n)*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-565e2f08268e5565
• 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-565e2f08268e5565
• 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-565e2f08268e5565

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-565e2f08268e5565

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

  • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-596b986e3a00db44 root: theorem check

      Parent-row provers of polytool polytool ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.