A132812 → D

OEIS matches ?

• A002740: 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:
  0:   0;
  1:   2;
  2:   9,     6;
  3:  24,    48,    12;
  4:  50,   200,   150,     20;
  5:  90,   600,   900,    360,     30;
  6: 147,  1470,  3675,   2940,    735,     42;
  7: 224,  3136, 11760,  15680,   7840,   1344,    56;
  8: 324,  6048, 31752,  63504,  52920,  18144,  2268,   72;
  9: 450, 10800, 75600, 211680, 264600, 151200, 37800, 3600, 90;

The row polynomials satisfy (4*n^4+2*n^3-10*n^2+4*n)*P(n,x) = -(52*n^4-52*n^3-22*n^2+22*n-(108*n^4-108*n^3-54*n^2+54*n)*x+(60*n^4-60*n^3-42*n^2+42*n)*x^2-(4*n^4-4*n^3-10*n^2+10*n)*x^3)*P(n-3,x) + (16*n^4-16*n^3-7*n^2+7*n-(48*n^4-48*n^3-24*n^2+24*n)*x+(48*n^4-48*n^3-30*n^2+30*n)*x^2-(16*n^4-16*n^3-16*n^2+16*n)*x^3-(3*n^2-3*n)*x^4)*d/dx P(n-3,x) - (92*n^4-46*n^3-44*n^2+22*n-(40*n^4-20*n^3-40*n^2+20*n)*x+(12*n^4-6*n^3-12*n^2+6*n)*x^2)*P(n-2,x) - (32*n^4-16*n^3-14*n^2+7*n+(6*n^2-3*n)*x-(32*n^4-16*n^3-14*n^2+7*n)*x^2-(6*n^2-3*n)*x^3)*d/dx P(n-2,x) + (20*n^4-28*n^2+8+(12*n^4-12*n^2)*x)*P(n-1,x) + (16*n^4-23*n^2+7-(16*n^4-26*n^2+10)*x-(3*n^2-3)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 9 + 6*x.

P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 9 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(52*n^4 - 52*n^3 - 22*n^2 + 22*n - (108*n^4 - 108*n^3 - 54*n^2 + 54*n)*x + (60*n^4 - 60*n^3 - 42*n^2 + 42*n)*x^2 - (4*n^4 - 4*n^3 - 10*n^2 + 10*n)*x^3)*P[n - 3, x] + (16*n^4 - 16*n^3 - 7*n^2 + 7*n - (48*n^4 - 48*n^3 - 24*n^2 + 24*n)*x + (48*n^4 - 48*n^3 - 30*n^2 + 30*n)*x^2 - (16*n^4 - 16*n^3 - 16*n^2 + 16*n)*x^3 - (3*n^2 - 3*n)*x^4)*D[P[n - 3, x], x] - (92*n^4 - 46*n^3 - 44*n^2 + 22*n - (40*n^4 - 20*n^3 - 40*n^2 + 20*n)*x + (12*n^4 - 6*n^3 - 12*n^2 + 6*n)*x^2)*P[n - 2, x] - (32*n^4 - 16*n^3 - 14*n^2 + 7*n + (6*n^2 - 3*n)*x - (32*n^4 - 16*n^3 - 14*n^2 + 7*n)*x^2 - (6*n^2 - 3*n)*x^3)*D[P[n - 2, x], x] + (20*n^4 - 28*n^2 + 8 + (12*n^4 - 12*n^2)*x)*P[n - 1, x] + (16*n^4 - 23*n^2 + 7 - (16*n^4 - 26*n^2 + 10)*x - (3*n^2 - 3)*x^2)*D[P[n - 1, x], x])/(4*n^4 + 2*n^3 - 10*n^2 + 4*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 9 + 6*x; for(n = 3, N, my(i = n + 1); v[i] = (-(52*n^4 - 52*n^3 - 22*n^2 + 22*n - (108*n^4 - 108*n^3 - 54*n^2 + 54*n)*x + (60*n^4 - 60*n^3 - 42*n^2 + 42*n)*x^2 - (4*n^4 - 4*n^3 - 10*n^2 + 10*n)*x^3)*v[i - 3] + (16*n^4 - 16*n^3 - 7*n^2 + 7*n - (48*n^4 - 48*n^3 - 24*n^2 + 24*n)*x + (48*n^4 - 48*n^3 - 30*n^2 + 30*n)*x^2 - (16*n^4 - 16*n^3 - 16*n^2 + 16*n)*x^3 - (3*n^2 - 3*n)*x^4)*deriv(v[i - 3], x) - (92*n^4 - 46*n^3 - 44*n^2 + 22*n - (40*n^4 - 20*n^3 - 40*n^2 + 20*n)*x + (12*n^4 - 6*n^3 - 12*n^2 + 6*n)*x^2)*v[i - 2] - (32*n^4 - 16*n^3 - 14*n^2 + 7*n + (6*n^2 - 3*n)*x - (32*n^4 - 16*n^3 - 14*n^2 + 7*n)*x^2 - (6*n^2 - 3*n)*x^3)*deriv(v[i - 2], x) + (20*n^4 - 28*n^2 + 8 + (12*n^4 - 12*n^2)*x)*v[i - 1] + (16*n^4 - 23*n^2 + 7 - (16*n^4 - 26*n^2 + 10)*x - (3*n^2 - 3)*x^2)*deriv(v[i - 1], x))/(4*n^4 + 2*n^3 - 10*n^2 + 4*n)); 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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-60496e4e2f616741
• Coefficients of one sign. Every nonzero row, n >= 0: 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-60496e4e2f616741
• Hadamard kernel (Malo). Rows P_n, n >= 0, 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-60496e4e2f616741

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-60496e4e2f616741

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05

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

      • premise (sequence): narayana
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check

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

        • preserver_image reversal, shift 0 e-29ef477df4172e22

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): narayana
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)
    • basis_map A007318 e-a32917d44ac940a7

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
        • preserver_image affine, shift 0 e-676bb32d6a1b9c64

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05 (expanded above)
            • preserver_image reversal, shift 0 e-fcd77baa2d5e1258

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05 (expanded above)
                • theorem_check liu_wang e-f8a41d41829f0ed1 root: theorem check

                  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

            • ↵ theorem_check liu_wang e-f8a41d41829f0ed1 (expanded above)
    • ↵ preserver_image reversal, shift 0 e-fcd77baa2d5e1258 (expanded above)
    • ↵ theorem_check liu_wang e-f8a41d41829f0ed1 (expanded above)

Related sequences ?

Same fingerprint (7)

Equal up to content, sign, reversal and a small index shift.

• A007318 → ⊙ C(n+1,k+1) (shift +2) → reverse
• A108838 → reverse
• A062196 → D
• A108838: Triangle of Dyck paths counted by number of long interior inclines.
• A352880: Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).
• narayana → D
• A001263 → D

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023