A011973 → x ↦ x−1/4 → (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;
  2:  3,   16;
  3:  4,   64;
  4:  5,  160,   144;
  5:  6,  320,   864;
  6:  7,  560,  3024,   1024;
  7:  8,  896,  8064,   8192;
  8:  9, 1344, 18144,  36864,  6400;
  9: 10, 1920, 36288, 122880, 64000;

The row polynomials satisfy (n-(10*n^3+2*n^2-4*n)*x+(4*n^5+8*n^4+4*n^3)*x^2)*P(n,x) = (n+1-(10*n^3+12*n^2-18*n+12)*x+(4*n^5+12*n^4-60*n^3-76*n^2+104*n+48)*x^2+(16*n^5+80*n^4+112*n^3-16*n^2-128*n-64)*x^3)*P(n-1,x) + (2*x-(20*n^2+44*n+24)*x^2+(8*n^4+48*n^3+184*n^2+272*n+96)*x^3-(32*n^4+192*n^3+416*n^2+384*n+128)*x^4)*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[((n + 1 - (10*n^3 + 12*n^2 - 18*n + 12)*x + (4*n^5 + 12*n^4 - 60*n^3 - 76*n^2 + 104*n + 48)*x^2 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^3)*P[n - 1, x] + (2*x - (20*n^2 + 44*n + 24)*x^2 + (8*n^4 + 48*n^3 + 184*n^2 + 272*n + 96)*x^3 - (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^4)*D[P[n - 1, x], x])/(n - (10*n^3 + 2*n^2 - 4*n)*x + (4*n^5 + 8*n^4 + 4*n^3)*x^2)]];
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] = ((n + 1 - (10*n^3 + 12*n^2 - 18*n + 12)*x + (4*n^5 + 12*n^4 - 60*n^3 - 76*n^2 + 104*n + 48)*x^2 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^3)*v[i - 1] + (2*x - (20*n^2 + 44*n + 24)*x^2 + (8*n^4 + 48*n^3 + 184*n^2 + 272*n + 96)*x^3 - (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^4)*deriv(v[i - 1], x))/(n - (10*n^3 + 2*n^2 - 4*n)*x + (4*n^5 + 8*n^4 + 4*n^3)*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-8f28dc5012800fde
• 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-8f28dc5012800fde
• 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-8f28dc5012800fde

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-8f28dc5012800fde

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

  • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1, shift 0 e-e2e7a7b682b47947

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

          Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

        • preserver_image reversal, shift 0 e-75f779345b39f910

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023