A086645 → ⊙ C(n+1,k) (shift +2)

OEIS matches ?

• A288470: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:
   1: 1,   12,      1;
   2: 1,   45,     45,        1;
   3: 1,  112,    420,      112,         1;
   4: 1,  225,   2100,     2100,       225,         1;
   5: 1,  396,   7425,    18480,      7425,       396,         1;
   6: 1,  637,  21021,   105105,    105105,     21021,       637,         1;
   7: 1,  960,  50960,   448448,    900900,    448448,     50960,       960,        1;
   8: 1, 1377, 110160,  1559376,   5513508,   5513508,   1559376,    110160,     1377,      1;
   9: 1, 1900, 218025,  4651200,  26453700,  46558512,  26453700,   4651200,   218025,   1900,    1;
  10: 1, 2541, 402325, 12311145, 105524100, 298750452, 298750452, 105524100, 12311145, 402325, 2541, 1;

The row polynomials satisfy (2*n^2+3*n+1)*P(n,x) = (30*n^2-11*n+1-(24*n^2-37*n-1)*x)*P(n-1,x) - ((54*n-48)*x-(54*n-48)*x^2)*d/dx P(n-1,x) - (36*n^2-34*n+12-(144*n^2-124*n+24)*x+(36*n^2+14*n-36)*x^2)*P(n-2,x) - (48*x-48*x^3)*d/dx P(n-2,x) + (8*n^2-20*n+12+(24*n^2-60*n+36)*x+(24*n^2-60*n+36)*x^2+(8*n^2-20*n+12)*x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1 + 12*x + x^2, P(2,x) = 1 + 45*x + 45*x^2 + x^3, P(3,x) = 1 + 112*x + 420*x^2 + 112*x^3 + x^4.

P[1, x_] := 1 + 12*x + x^2;
P[2, x_] := 1 + 45*x + 45*x^2 + x^3;
P[3, x_] := 1 + 112*x + 420*x^2 + 112*x^3 + x^4;
P[n_, x_] := P[n, x] = Expand[Together[((30*n^2 - 11*n + 1 - (24*n^2 - 37*n - 1)*x)*P[n - 1, x] - ((54*n - 48)*x - (54*n - 48)*x^2)*D[P[n - 1, x], x] - (36*n^2 - 34*n + 12 - (144*n^2 - 124*n + 24)*x + (36*n^2 + 14*n - 36)*x^2)*P[n - 2, x] - (48*x - 48*x^3)*D[P[n - 2, x], x] + (8*n^2 - 20*n + 12 + (24*n^2 - 60*n + 36)*x + (24*n^2 - 60*n + 36)*x^2 + (8*n^2 - 20*n + 12)*x^3)*P[n - 3, x])/(2*n^2 + 3*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 12*x + x^2; v[2] = 1 + 45*x + 45*x^2 + x^3; v[3] = 1 + 112*x + 420*x^2 + 112*x^3 + x^4; for(n = 4, N, my(i = n); v[i] = ((30*n^2 - 11*n + 1 - (24*n^2 - 37*n - 1)*x)*v[i - 1] - ((54*n - 48)*x - (54*n - 48)*x^2)*deriv(v[i - 1], x) - (36*n^2 - 34*n + 12 - (144*n^2 - 124*n + 24)*x + (36*n^2 + 14*n - 36)*x^2)*v[i - 2] - (48*x - 48*x^3)*deriv(v[i - 2], x) + (8*n^2 - 20*n + 12 + (24*n^2 - 60*n + 36)*x + (24*n^2 - 60*n + 36)*x^2 + (8*n^2 - 20*n + 12)*x^3)*v[i - 3])/(2*n^2 + 3*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-e37e9c5997239324
• Coefficients of one sign. Every nonzero row, n >= 1: 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-e37e9c5997239324
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-e37e9c5997239324

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 hadamard_multiplier, shift 2 e-e37e9c5997239324

  Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
    uses: Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-1375f2f1a11f9809 root: theorem check

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

    • preserver_image reversal, shift 0 e-a5e3cd5a8b6d6a1a

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-1375f2f1a11f9809 (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