A060920 → (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,     4;
  2:    5,    20,      9;
  3:   13,    80,     81,     16;
  4:   34,   284,    459,    224,     25;
  5:   89,   940,   2097,   1680,    500,     36;
  6:  233,  2976,   8478,   9504,   4750,    972,     49;
  7:  610,  9140,  31698,  45760,  32375,  11340,   1715,    64;
  8: 1597, 27460, 112257, 198432, 182125,  91224,  24010,  2816,   81;
  9: 4181, 81136, 382023, 799408, 903050, 590652, 224322, 46464, 4374, 100;

The row polynomials satisfy (200*n-100+(500*n^2-330*n+40)*x+(600*n^3-740*n^2+308*n-44)*x^2+(240*n^4-280*n^3+28*n^2+34*n-4)*x^3+(32*n^5-48*n^4+24*n^3-4*n^2)*x^4)*P(n,x) = (800*n-600+(2700*n^2-1170*n-710)*x+(3000*n^3+700*n^2-2788*n+176)*x^2+(1120*n^4+2440*n^3-2288*n^2-1306*n+566)*x^3+(128*n^5+976*n^4-160*n^3-956*n^2-116*n+208)*x^4+(96*n^5+80*n^4-152*n^3-52*n^2+56*n-12)*x^5)*P(n-1,x) - (125+(750*n+150)*x+(1200*n^2+1650*n-240)*x^2+(600*n^3+2660*n^2+1110*n-524)*x^3+(80*n^4+1240*n^3+1880*n^2+146*n-325)*x^4+(144*n^4+768*n^3+436*n^2-96*n-58)*x^5+(64*n^4+128*n^3+16*n^2-32*n+8)*x^6)*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[((800*n - 600 + (2700*n^2 - 1170*n - 710)*x + (3000*n^3 + 700*n^2 - 2788*n + 176)*x^2 + (1120*n^4 + 2440*n^3 - 2288*n^2 - 1306*n + 566)*x^3 + (128*n^5 + 976*n^4 - 160*n^3 - 956*n^2 - 116*n + 208)*x^4 + (96*n^5 + 80*n^4 - 152*n^3 - 52*n^2 + 56*n - 12)*x^5)*P[n - 1, x] - (125 + (750*n + 150)*x + (1200*n^2 + 1650*n - 240)*x^2 + (600*n^3 + 2660*n^2 + 1110*n - 524)*x^3 + (80*n^4 + 1240*n^3 + 1880*n^2 + 146*n - 325)*x^4 + (144*n^4 + 768*n^3 + 436*n^2 - 96*n - 58)*x^5 + (64*n^4 + 128*n^3 + 16*n^2 - 32*n + 8)*x^6)*D[P[n - 1, x], x])/(200*n - 100 + (500*n^2 - 330*n + 40)*x + (600*n^3 - 740*n^2 + 308*n - 44)*x^2 + (240*n^4 - 280*n^3 + 28*n^2 + 34*n - 4)*x^3 + (32*n^5 - 48*n^4 + 24*n^3 - 4*n^2)*x^4)]];
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] = ((800*n - 600 + (2700*n^2 - 1170*n - 710)*x + (3000*n^3 + 700*n^2 - 2788*n + 176)*x^2 + (1120*n^4 + 2440*n^3 - 2288*n^2 - 1306*n + 566)*x^3 + (128*n^5 + 976*n^4 - 160*n^3 - 956*n^2 - 116*n + 208)*x^4 + (96*n^5 + 80*n^4 - 152*n^3 - 52*n^2 + 56*n - 12)*x^5)*v[i - 1] - (125 + (750*n + 150)*x + (1200*n^2 + 1650*n - 240)*x^2 + (600*n^3 + 2660*n^2 + 1110*n - 524)*x^3 + (80*n^4 + 1240*n^3 + 1880*n^2 + 146*n - 325)*x^4 + (144*n^4 + 768*n^3 + 436*n^2 - 96*n - 58)*x^5 + (64*n^4 + 128*n^3 + 16*n^2 - 32*n + 8)*x^6)*deriv(v[i - 1], x))/(200*n - 100 + (500*n^2 - 330*n + 40)*x + (600*n^3 - 740*n^2 + 308*n - 44)*x^2 + (240*n^4 - 280*n^3 + 28*n^2 + 34*n - 4)*x^3 + (32*n^5 - 48*n^4 + 24*n^3 - 4*n^2)*x^4)); 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-dda6d332e97688a5

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-dda6d332e97688a5

  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