A113413 → θ+1

OEIS matches ?

• A119915: 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: 1;
  1: 2,  2;
  2: 2,  8,   3;
  3: 2, 16,  18,    4;
  4: 2, 24,  54,   32,    5;
  5: 2, 32, 114,  128,   50,    6;
  6: 2, 40, 198,  352,  250,   72,    7;
  7: 2, 48, 306,  768,  850,  432,   98,    8;
  8: 2, 56, 438, 1440, 2250, 1752,  686,  128,   9;
  9: 2, 64, 594, 2432, 5010, 5472, 3234, 1024, 162, 10;

The row polynomials satisfy -(n^2-n+(2*n^3-5*n^2-n)*x-(2*n^3+5*n^2-n)*x^2+(n^2+n)*x^3)*P(n,x) = -(n^2-n+(2*n^3+3*n^2-13*n)*x+(6*n^3-11*n^2-23*n+8)*x^2-(9*n^2+5*n-6)*x^3+(2*n+2)*x^4)*P(n-1,x) + (n*x+(2*n^2+8*n-1)*x^2+(12*n^2+12*n-7)*x^3+(2*n^2-4*n-7)*x^4-(n+1)*x^5)*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 - n + (2*n^3 + 3*n^2 - 13*n)*x + (6*n^3 - 11*n^2 - 23*n + 8)*x^2 - (9*n^2 + 5*n - 6)*x^3 + (2*n + 2)*x^4)*P[n - 1, x] + (n*x + (2*n^2 + 8*n - 1)*x^2 + (12*n^2 + 12*n - 7)*x^3 + (2*n^2 - 4*n - 7)*x^4 - (n + 1)*x^5)*D[P[n - 1, x], x])/(-n^2 + n - (2*n^3 - 5*n^2 - n)*x + (2*n^3 + 5*n^2 - n)*x^2 - (n^2 + n)*x^3)]];
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 - n + (2*n^3 + 3*n^2 - 13*n)*x + (6*n^3 - 11*n^2 - 23*n + 8)*x^2 - (9*n^2 + 5*n - 6)*x^3 + (2*n + 2)*x^4)*v[i - 1] + (n*x + (2*n^2 + 8*n - 1)*x^2 + (12*n^2 + 12*n - 7)*x^3 + (2*n^2 - 4*n - 7)*x^4 - (n + 1)*x^5)*deriv(v[i - 1], x))/(-n^2 + n - (2*n^3 - 5*n^2 - n)*x + (2*n^3 + 5*n^2 - n)*x^2 - (n^2 + n)*x^3)); 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-ef030cea3fd1d83e
• 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-ef030cea3fd1d83e
• 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-ef030cea3fd1d83e
• 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-ef030cea3fd1d83e

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-ef030cea3fd1d83e

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

  • premise (sequence): A Riordan array of coordination sequences.
    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-7fe1799d89540b32 root: theorem check

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

Related sequences ?

Same fingerprint (1)

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

• A122542 → 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. [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.