A116412 → (θ+1)²

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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:   3,     4;
  2:   6,    24,      9;
  3:  12,    84,     81,     16;
  4:  24,   240,    405,    192,     25;
  5:  48,   624,   1539,   1248,    375,     36;
  6:  96,  1536,   5022,   5952,   3000,    648,    49;
  7: 192,  3648,  14904,  23568,  17250,   6156,  1029,    64;
  8: 384,  8448,  41472,  82560,  80625,  41472, 11319,  1536,   81;
  9: 768, 19200, 110160, 265344, 327075, 223884, 87465, 19200, 2187, 100;

The row polynomials satisfy (16*n^2-32*n+16+(36*n^3-92*n^2+48*n)*x+(9*n^4-18*n^3-33*n^2+42*n)*x^2-(9*n^3-4*n^2-7*n+2)*x^3+n^2*x^4)*P(n,x) = (16*n^2*x+(36*n^3+16*n^2-28*n-8)*x^2+(9*n^4+18*n^3-33*n^2-42*n)*x^3-(9*n^3+23*n^2+12*n)*x^4+(n^2+2*n+1)*x^5)*P(n-2,x) + (32*n^2-64*n+32+(72*n^3-96*n^2-96*n+96)*x+(18*n^4+36*n^3-194*n^2-52*n+160)*x^2+(9*n^4-18*n^3-97*n^2+26*n+80)*x^3-(9*n^3+12*n^2-12*n-12)*x^4+(n^2+2*n+1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2*x + (36*n^3 + 16*n^2 - 28*n - 8)*x^2 + (9*n^4 + 18*n^3 - 33*n^2 - 42*n)*x^3 - (9*n^3 + 23*n^2 + 12*n)*x^4 + (n^2 + 2*n + 1)*x^5)*P[n - 2, x] + (32*n^2 - 64*n + 32 + (72*n^3 - 96*n^2 - 96*n + 96)*x + (18*n^4 + 36*n^3 - 194*n^2 - 52*n + 160)*x^2 + (9*n^4 - 18*n^3 - 97*n^2 + 26*n + 80)*x^3 - (9*n^3 + 12*n^2 - 12*n - 12)*x^4 + (n^2 + 2*n + 1)*x^5)*P[n - 1, x])/(16*n^2 - 32*n + 16 + (36*n^3 - 92*n^2 + 48*n)*x + (9*n^4 - 18*n^3 - 33*n^2 + 42*n)*x^2 - (9*n^3 - 4*n^2 - 7*n + 2)*x^3 + 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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^2*x + (36*n^3 + 16*n^2 - 28*n - 8)*x^2 + (9*n^4 + 18*n^3 - 33*n^2 - 42*n)*x^3 - (9*n^3 + 23*n^2 + 12*n)*x^4 + (n^2 + 2*n + 1)*x^5)*v[i - 2] + (32*n^2 - 64*n + 32 + (72*n^3 - 96*n^2 - 96*n + 96)*x + (18*n^4 + 36*n^3 - 194*n^2 - 52*n + 160)*x^2 + (9*n^4 - 18*n^3 - 97*n^2 + 26*n + 80)*x^3 - (9*n^3 + 12*n^2 - 12*n - 12)*x^4 + (n^2 + 2*n + 1)*x^5)*v[i - 1])/(16*n^2 - 32*n + 16 + (36*n^3 - 92*n^2 + 48*n)*x + (9*n^4 - 18*n^3 - 33*n^2 + 42*n)*x^2 - (9*n^3 - 4*n^2 - 7*n + 2)*x^3 + 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-8755b73a3425b06c
• 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-8755b73a3425b06c
• 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-8755b73a3425b06c
• 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-8755b73a3425b06c

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-8755b73a3425b06c

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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 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 (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.