A135838 → (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:  2,    8;
  2:  2,   16,    18;
  3:  4,   48,   108,    64;
  4:  4,   64,   216,   256,    100;
  5:  8,  160,   720,  1280,   1000,    288;
  6:  8,  192,  1080,  2560,   3000,   1728,    392;
  7: 16,  448,  3024,  8960,  14000,  12096,   5488,  1024;
  8: 16,  512,  4032, 14336,  28000,  32256,  21952,  8192,  1296;
  9: 32, 1152, 10368, 43008, 100800, 145152, 131712, 73728, 23328, 3200;

The row polynomials satisfy (1+(3*n-4)*x+(n^2-2*n+1)*x^2)*P(n,x) = (2+(6*n+8)*x+(2*n^2+16*n+12)*x^2+(4*n^2+14*n+8)*x^3+(2*n^2+4*n+2)*x^4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n + 8)*x + (2*n^2 + 16*n + 12)*x^2 + (4*n^2 + 14*n + 8)*x^3 + (2*n^2 + 4*n + 2)*x^4)*P[n - 2, x])/(1 + (3*n - 4)*x + (n^2 - 2*n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((2 + (6*n + 8)*x + (2*n^2 + 16*n + 12)*x^2 + (4*n^2 + 14*n + 8)*x^3 + (2*n^2 + 4*n + 2)*x^4)*v[i - 2])/(1 + (3*n - 4)*x + (n^2 - 2*n + 1)*x^2)); 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-064e625d7496a635
• 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-064e625d7496a635
• 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-064e625d7496a635

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-064e625d7496a635

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

  • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-07f7d31230240b98

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-51cb02553cf4a8ac root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-51cb02553cf4a8ac (expanded above)

Related sequences ?

Same fingerprint (12)

Equal up to content, sign, reversal and a small index shift.

• basis-map coefficients of A113214, shifted → (θ+1)²
• A344565 → (θ+1)²
• A038208 → (θ+1)²
• A007318 → ×(1+2n+n²) → (θ+1)²
• A028326 → (θ+1)²
• A007318 → (θ+1)² → reverse
• A085880 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A007318 → D → (θ+1)²
• A178821 → (θ+1)²
• A178820 → (θ+1)²
• A178822 → (θ+1)²

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.