A172431 → ⊙ C(n,k+1) (shift -1) → x ↦ x+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:
   1:      1;
   2:      2,      1;
   3:      6,      6,       1;
   4:     21,     32,      13,       1;
   5:     80,    164,     108,      24,      1;
   6:    322,    830,     760,     290,     40,      1;
   7:   1347,   4184,    4918,    2692,    673,     62,     1;
   8:   5798,  21077,   30282,   21700,   8015,   1407,    91,    1;
   9:  25512, 106248,  180516,  160272,  78960,  21056,  2716,  128,   1;
  10: 114236, 536256, 1052100, 1115676, 689904, 249804, 50244, 4920, 174, 1;

The row polynomials satisfy (4*n^3-2*n^2-2*n)*P(n,x) = (40*n^3-118*n^2+108*n-24+(36*n^3-116*n^2+110*n-24)*x)*P(n-1,x) - (44*n^2-98*n+30+(96*n^2-216*n+66)*x+(52*n^2-118*n+36)*x^2)*d/dx P(n-1,x) + (15*n-6+(50*n-20)*x+(55*n-22)*x^2+(20*n-8)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((40*n^3 - 118*n^2 + 108*n - 24 + (36*n^3 - 116*n^2 + 110*n - 24)*x)*P[n - 1, x] - (44*n^2 - 98*n + 30 + (96*n^2 - 216*n + 66)*x + (52*n^2 - 118*n + 36)*x^2)*D[P[n - 1, x], x] + (15*n - 6 + (50*n - 20)*x + (55*n - 22)*x^2 + (20*n - 8)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 2*n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((40*n^3 - 118*n^2 + 108*n - 24 + (36*n^3 - 116*n^2 + 110*n - 24)*x)*v[i - 1] - (44*n^2 - 98*n + 30 + (96*n^2 - 216*n + 66)*x + (52*n^2 - 118*n + 36)*x^2)*deriv(v[i - 1], x) + (15*n - 6 + (50*n - 20)*x + (55*n - 22)*x^2 + (20*n - 8)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 2*n^2 - 2*n)); 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, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-9b9e29c7199cdb19
• 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-9b9e29c7199cdb19
• 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-9b9e29c7199cdb19

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 affine, shift 0 e-9b9e29c7199cdb19

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image hadamard_multiplier, shift -1 e-1491609a6fc5f8c9

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

      • premise (sequence): Even row Pascal-square read by antidiagonals.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-83282836b5ac538e root: theorem check

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

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