A060920 → 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:    2,     2;
  2:    5,    10,      3;
  3:   13,    40,     27,      4;
  4:   34,   142,    153,     56,      5;
  5:   89,   470,    699,    420,    100,     6;
  6:  233,  1488,   2826,   2376,    950,   162,     7;
  7:  610,  4570,  10566,  11440,   6475,  1890,   245,    8;
  8: 1597, 13730,  37419,  49608,  36425, 15204,  3430,  352,   9;
  9: 4181, 40568, 127341, 199852, 180610, 98442, 32046, 5808, 486, 10;

The row polynomials satisfy (20*n-10+(20*n^2-6*n-2)*x+(8*n^3-8*n^2+2*n)*x^2)*P(n,x) = (80*n-60+(100*n^2+26*n-82)*x+(32*n^3+80*n^2-64*n-20)*x^2+(24*n^3-26*n+6)*x^3)*P(n-1,x) - (25+(50*n+55)*x+(20*n^2+110*n+33)*x^2+(36*n^2+76*n-1)*x^3+(16*n^2+16*n-4)*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[((80*n - 60 + (100*n^2 + 26*n - 82)*x + (32*n^3 + 80*n^2 - 64*n - 20)*x^2 + (24*n^3 - 26*n + 6)*x^3)*P[n - 1, x] - (25 + (50*n + 55)*x + (20*n^2 + 110*n + 33)*x^2 + (36*n^2 + 76*n - 1)*x^3 + (16*n^2 + 16*n - 4)*x^4)*D[P[n - 1, x], x])/(20*n - 10 + (20*n^2 - 6*n - 2)*x + (8*n^3 - 8*n^2 + 2*n)*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] = ((80*n - 60 + (100*n^2 + 26*n - 82)*x + (32*n^3 + 80*n^2 - 64*n - 20)*x^2 + (24*n^3 - 26*n + 6)*x^3)*v[i - 1] - (25 + (50*n + 55)*x + (20*n^2 + 110*n + 33)*x^2 + (36*n^2 + 76*n - 1)*x^3 + (16*n^2 + 16*n - 4)*x^4)*deriv(v[i - 1], x))/(20*n - 10 + (20*n^2 - 6*n - 2)*x + (8*n^3 - 8*n^2 + 2*n)*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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-2b404cb8e7cd55aa

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-2b404cb8e7cd55aa

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

  • premise (sequence): Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zero…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check three_term_recurrence e-1b409596bf82290e root: theorem check

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

    • preserver_image affine, shift 0 e-f409e7a544612dfe

      Affine, reversal and Möbius substitutions (elementary) 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
    • basis_map A007318 e-5dc8ce8b8b471ab9

      Geometric basis map (elementary) 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 (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-313e6d9276f9a630 (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