A059473 → θ+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:
  0:   1;
  1:   2,     4;
  2:   4,    20,     12;
  3:   8,    64,     96,     32;
  4:  16,   176,    444,    352,     80;
  5:  32,   448,   1608,   2144,   1120,     192;
  6:  64,  1088,   5088,   9760,   8480,    3264,    448;
  7: 128,  2560,  14784,  37376,  46720,   29568,   8960,   1024;
  8: 256,  5888,  40512, 127744, 211280,  191616,  94528,  23552,  2304;
  9: 512, 13312, 106368, 402944, 835360, 1002432, 705152, 283648, 59904, 5120;

The row polynomials satisfy -(2*n+(3*n^2-4*n)*x-(3*n^2+4*n)*x^2)*P(n,x) = -(4*n+(6*n^2+16*n-16)*x+(12*n^2-2*n-28)*x^2-(12*n+16)*x^3)*P(n-1,x) + (4*x+(6*n+24)*x^2+(24*n+36)*x^3+(6*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[(-(4*n + (6*n^2 + 16*n - 16)*x + (12*n^2 - 2*n - 28)*x^2 - (12*n + 16)*x^3)*P[n - 1, x] + (4*x + (6*n + 24)*x^2 + (24*n + 36)*x^3 + (6*n + 8)*x^4)*D[P[n - 1, x], x])/(-2*n - (3*n^2 - 4*n)*x + (3*n^2 + 4*n)*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] = (-(4*n + (6*n^2 + 16*n - 16)*x + (12*n^2 - 2*n - 28)*x^2 - (12*n + 16)*x^3)*v[i - 1] + (4*x + (6*n + 24)*x^2 + (24*n + 36)*x^3 + (6*n + 8)*x^4)*deriv(v[i - 1], x))/(-2*n - (3*n^2 - 4*n)*x + (3*n^2 + 4*n)*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-6cc6a3c9200dfa14
• 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-6cc6a3c9200dfa14
• 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-6cc6a3c9200dfa14
• 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-6cc6a3c9200dfa14

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-6cc6a3c9200dfa14

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

  • premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
    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-0ee228386582da48 root: theorem check

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

    • preserver_image reversal, shift 0 e-c68c82cb1005adf9

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
        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-0ee228386582da48 (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.