A060920 → D+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:     3,      1;
  2:    10,      7,      1;
  3:    33,     38,     12,      1;
  4:   105,    173,     93,     18,      1;
  5:   324,    701,    548,    185,     25,     1;
  6:   977,   2628,   2724,   1354,    325,    33,    1;
  7:  2895,   9329,  12102,   8040,   2870,   525,   42,    1;
  8:  8462,  31811,  49679,  41542,  19955,  5474,  798,   52,  1;
  9: 24465, 105178, 192336, 194451, 118157, 43875, 9660, 1158, 63, 1;

The row polynomials satisfy (8*n^3-4*n^2-10*n+5+(16*n^2-18*n+5)*x+(8*n-4)*x^2)*P(n,x) = (32*n^3-66*n+31+(24*n^3+60*n^2-108*n+32)*x+(48*n^2-14*n-19)*x^2+(24*n-20)*x^3)*P(n-1,x) - (20*n^2+30*n-5+(36*n^2+94*n+6)*x+(16*n^2+96*n+43)*x^2+(32*n+48)*x^3+16*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[((32*n^3 - 66*n + 31 + (24*n^3 + 60*n^2 - 108*n + 32)*x + (48*n^2 - 14*n - 19)*x^2 + (24*n - 20)*x^3)*P[n - 1, x] - (20*n^2 + 30*n - 5 + (36*n^2 + 94*n + 6)*x + (16*n^2 + 96*n + 43)*x^2 + (32*n + 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 4*n^2 - 10*n + 5 + (16*n^2 - 18*n + 5)*x + (8*n - 4)*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] = ((32*n^3 - 66*n + 31 + (24*n^3 + 60*n^2 - 108*n + 32)*x + (48*n^2 - 14*n - 19)*x^2 + (24*n - 20)*x^3)*v[i - 1] - (20*n^2 + 30*n - 5 + (36*n^2 + 94*n + 6)*x + (16*n^2 + 96*n + 43)*x^2 + (32*n + 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 4*n^2 - 10*n + 5 + (16*n^2 - 18*n + 5)*x + (8*n - 4)*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-e3271f37ad24627f

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 derivative_polynomial, shift 0 e-e3271f37ad24627f

  Laguerre’s theorem on differential operators [SP14] 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023