A249251 → (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: 1,  12;
  2: 1,  24,   54;
  3: 1,  36,  189,   192;
  4: 1,  48,  405,   960,    600;
  5: 1,  60,  702,  2736,   3900,   1728;
  6: 1,  72, 1080,  5952,  13950,  13824,   4704;
  7: 1,  84, 1539, 11040,  36825,  59616,  44688,  12288;
  8: 1,  96, 2079, 18432,  80625, 185760, 225792, 135168,  31104;
  9: 1, 108, 2700, 28560, 155475, 470988, 812616, 783360, 388800, 76800;

The row polynomials satisfy (n^2+2*n+1+(9*n^3+19*n^2+12*n+4)*x+(9*n^4+30*n^3+51*n^2+54*n+8)*x^2+(12*n^3+40*n^2+44*n-16)*x^3+16*n^2*x^4)*P(n,x) = ((n^2+4*n+4)*x+(9*n^3+46*n^2+77*n+44)*x^2+(9*n^4+66*n^3+195*n^2+282*n+152)*x^3+(12*n^3+76*n^2+160*n+80)*x^4+(16*n^2+32*n+16)*x^5)*P(n-2,x) + (n^2+2*n+1+(9*n^3+30*n^2+32*n+12)*x+(9*n^4+66*n^3+155*n^2+178*n+96)*x^2+(18*n^4+108*n^3+310*n^2+476*n+288)*x^3+(24*n^3+144*n^2+320*n+160)*x^4+(32*n^2+64*n+32)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.

P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(((n^2 + 4*n + 4)*x + (9*n^3 + 46*n^2 + 77*n + 44)*x^2 + (9*n^4 + 66*n^3 + 195*n^2 + 282*n + 152)*x^3 + (12*n^3 + 76*n^2 + 160*n + 80)*x^4 + (16*n^2 + 32*n + 16)*x^5)*P[n - 2, x] + (n^2 + 2*n + 1 + (9*n^3 + 30*n^2 + 32*n + 12)*x + (9*n^4 + 66*n^3 + 155*n^2 + 178*n + 96)*x^2 + (18*n^4 + 108*n^3 + 310*n^2 + 476*n + 288)*x^3 + (24*n^3 + 144*n^2 + 320*n + 160)*x^4 + (32*n^2 + 64*n + 32)*x^5)*P[n - 1, x])/(n^2 + 2*n + 1 + (9*n^3 + 19*n^2 + 12*n + 4)*x + (9*n^4 + 30*n^3 + 51*n^2 + 54*n + 8)*x^2 + (12*n^3 + 40*n^2 + 44*n - 16)*x^3 + 16*n^2*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 + 4*n + 4)*x + (9*n^3 + 46*n^2 + 77*n + 44)*x^2 + (9*n^4 + 66*n^3 + 195*n^2 + 282*n + 152)*x^3 + (12*n^3 + 76*n^2 + 160*n + 80)*x^4 + (16*n^2 + 32*n + 16)*x^5)*v[i - 2] + (n^2 + 2*n + 1 + (9*n^3 + 30*n^2 + 32*n + 12)*x + (9*n^4 + 66*n^3 + 155*n^2 + 178*n + 96)*x^2 + (18*n^4 + 108*n^3 + 310*n^2 + 476*n + 288)*x^3 + (24*n^3 + 144*n^2 + 320*n + 160)*x^4 + (32*n^2 + 64*n + 32)*x^5)*v[i - 1])/(n^2 + 2*n + 1 + (9*n^3 + 19*n^2 + 12*n + 4)*x + (9*n^4 + 30*n^3 + 51*n^2 + 54*n + 8)*x^2 + (12*n^3 + 40*n^2 + 44*n - 16)*x^3 + 16*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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-b2da355999d9253c
• 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-b2da355999d9253c
• 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-b2da355999d9253c
• 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-b2da355999d9253c

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-b2da355999d9253c

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

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-67b682ca6511d839 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

    • preserver_image reversal, shift 0 e-c7b023a4533e2987

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang_derivative_lag e-ee40b91a84fdd5b7 root: theorem check

          Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

        • preserver_image reversal, shift 0 e-c36241f3b645d7b5

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang_derivative_lag e-67b682ca6511d839 (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.