A145596 → (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,    8;
  2:  3,   32,    27;
  3:  4,   80,   180,     64;
  4:  5,  160,   675,    640,    125;
  5:  6,  280,  1890,   3360,   1750,    216;
  6:  7,  448,  4410,  12544,  12250,   4032,    343;
  7:  8,  672,  9072,  37632,  58800,  36288,   8232,    512;
  8:  9,  960, 17010,  96768, 220500, 217728,  92610,  15360,   729;
  9: 10, 1320, 29700, 221760, 693000, 997920, 679140, 211200, 26730, 1000;

The row polynomials satisfy (n^2+4*n+(2*n^4+10*n^3+6*n^2-8*n)*x+(n^6+8*n^5+20*n^4+16*n^3)*x^2)*P(n,x) = (n^2+5*n+4+(2*n^4+12*n^3+28*n^2+18*n)*x+(n^6+9*n^5+37*n^4+69*n^3+47*n^2+7*n)*x^2+(3*n^6+25*n^5+77*n^4+113*n^3+92*n^2+42*n+8)*x^3)*P(n-1,x) + ((2*n+2)*x+(4*n^3+14*n^2+10*n)*x^2+(2*n^5+16*n^4+42*n^3+44*n^2+22*n+6)*x^3-(2*n^5+16*n^4+46*n^3+58*n^2+34*n+8)*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[((n^2 + 5*n + 4 + (2*n^4 + 12*n^3 + 28*n^2 + 18*n)*x + (n^6 + 9*n^5 + 37*n^4 + 69*n^3 + 47*n^2 + 7*n)*x^2 + (3*n^6 + 25*n^5 + 77*n^4 + 113*n^3 + 92*n^2 + 42*n + 8)*x^3)*P[n - 1, x] + ((2*n + 2)*x + (4*n^3 + 14*n^2 + 10*n)*x^2 + (2*n^5 + 16*n^4 + 42*n^3 + 44*n^2 + 22*n + 6)*x^3 - (2*n^5 + 16*n^4 + 46*n^3 + 58*n^2 + 34*n + 8)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + (2*n^4 + 10*n^3 + 6*n^2 - 8*n)*x + (n^6 + 8*n^5 + 20*n^4 + 16*n^3)*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] = ((n^2 + 5*n + 4 + (2*n^4 + 12*n^3 + 28*n^2 + 18*n)*x + (n^6 + 9*n^5 + 37*n^4 + 69*n^3 + 47*n^2 + 7*n)*x^2 + (3*n^6 + 25*n^5 + 77*n^4 + 113*n^3 + 92*n^2 + 42*n + 8)*x^3)*v[i - 1] + ((2*n + 2)*x + (4*n^3 + 14*n^2 + 10*n)*x^2 + (2*n^5 + 16*n^4 + 42*n^3 + 44*n^2 + 22*n + 6)*x^3 - (2*n^5 + 16*n^4 + 46*n^3 + 58*n^2 + 34*n + 8)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + (2*n^4 + 10*n^3 + 6*n^2 - 8*n)*x + (n^6 + 8*n^5 + 20*n^4 + 16*n^3)*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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-915e0a73a4c39c54
• 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-915e0a73a4c39c54
• 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-915e0a73a4c39c54
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-915e0a73a4c39c54

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-915e0a73a4c39c54

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

  • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c483eeee9c2638c9 root: theorem check

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

    • preserver_image reversal, shift 0 e-1d2c7b4a3cb3d72f

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-c483eeee9c2638c9 (expanded above)
        • basis_map A007318 e-fb63394f9047a486

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
            • preserver_image affine, shift 0 e-3babd9179153850c

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-c483eeee9c2638c9 (expanded above)
    • ↵ basis_map A007318 e-fb63394f9047a486 (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. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.