A054142 → (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, 12,    9;
  3: 1, 20,   54,   16;
  4: 1, 28,  135,  160,    25;
  5: 1, 36,  252,  560,   375,    36;
  6: 1, 44,  405, 1344,  1750,   756,    49;
  7: 1, 52,  594, 2640,  5250,  4536,  1372,    64;
  8: 1, 60,  819, 4576, 12375, 16632, 10290,  2304,   81;
  9: 1, 68, 1080, 7280, 25025, 46332, 45276, 21120, 3645, 100;

The row polynomials satisfy (8*n^3-4*n^2+(48*n^4-72*n^3+28*n^2-2*n)*x+(32*n^5-48*n^4+24*n^3-4*n^2)*x^2)*P(n,x) = (8*n^3-4*n^2+(48*n^4+24*n^3-52*n^2-2*n)*x+(32*n^5+192*n^4-120*n^3-152*n^2+48*n+12)*x^2+(96*n^5+80*n^4-152*n^3-52*n^2+56*n-12)*x^3)*P(n-1,x) - (4*n^2*x+(24*n^3+28*n^2-6*n-1)*x^2+(16*n^4+128*n^3+52*n^2-32*n-2)*x^3+(64*n^4+128*n^3+16*n^2-32*n+8)*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[((8*n^3 - 4*n^2 + (48*n^4 + 24*n^3 - 52*n^2 - 2*n)*x + (32*n^5 + 192*n^4 - 120*n^3 - 152*n^2 + 48*n + 12)*x^2 + (96*n^5 + 80*n^4 - 152*n^3 - 52*n^2 + 56*n - 12)*x^3)*P[n - 1, x] - (4*n^2*x + (24*n^3 + 28*n^2 - 6*n - 1)*x^2 + (16*n^4 + 128*n^3 + 52*n^2 - 32*n - 2)*x^3 + (64*n^4 + 128*n^3 + 16*n^2 - 32*n + 8)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 4*n^2 + (48*n^4 - 72*n^3 + 28*n^2 - 2*n)*x + (32*n^5 - 48*n^4 + 24*n^3 - 4*n^2)*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] = ((8*n^3 - 4*n^2 + (48*n^4 + 24*n^3 - 52*n^2 - 2*n)*x + (32*n^5 + 192*n^4 - 120*n^3 - 152*n^2 + 48*n + 12)*x^2 + (96*n^5 + 80*n^4 - 152*n^3 - 52*n^2 + 56*n - 12)*x^3)*v[i - 1] - (4*n^2*x + (24*n^3 + 28*n^2 - 6*n - 1)*x^2 + (16*n^4 + 128*n^3 + 52*n^2 - 32*n - 2)*x^3 + (64*n^4 + 128*n^3 + 16*n^2 - 32*n + 8)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 4*n^2 + (48*n^4 - 72*n^3 + 28*n^2 - 2*n)*x + (32*n^5 - 48*n^4 + 24*n^3 - 4*n^2)*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-34cb5f7197630da3
• 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-34cb5f7197630da3
• 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-34cb5f7197630da3
• 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-34cb5f7197630da3

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-34cb5f7197630da3

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

  • premise (sequence): Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-abbe6e2e6a5c7279 root: theorem check

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

    • gamma_polynomial scale 1, shift 0 e-313e6d9276f9a630

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • basis_map A008288 e-80180a24956ab79f

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-a8f2d1946d8f3a6f

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-054f97e0823e3ab6

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-912230b718f958a6

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement

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