A173018 → D

OEIS matches ?

• A142706: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A001286: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 2 ?
• A180119: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:
   1:   0;
   2:   0;
   3:   1;
   4:   4,     2;
   5:  11,    22,      3;
   6:  26,   132,     78,      4;
   7:  57,   604,    906,    228,      5;
   8: 120,  2382,   7248,   4764,    600,     6;
   9: 247,  8586,  46857,  62476,  21465,  1482,    7;
  10: 502, 29216, 264702, 624760, 441170, 87648, 3514, 8;

The row polynomials satisfy (n-3)*P(n,x) = -(2*n-4+(2*n^2-12*n+16)*x+(n^3-9*n^2+26*n-24)*x^2)*P(n-2,x) - ((n-2)*x+(n^2-6*n+8)*x^2-(n^2-5*n+6)*x^3)*d/dx P(n-2,x) + (3*n-8+(2*n^2-12*n+18)*x)*P(n-1,x) + ((n-3)*x-(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.

P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (2*n^2 - 12*n + 16)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*P[n - 2, x] - ((n - 2)*x + (n^2 - 6*n + 8)*x^2 - (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] + (3*n - 8 + (2*n^2 - 12*n + 18)*x)*P[n - 1, x] + ((n - 3)*x - (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = (-(2*n - 4 + (2*n^2 - 12*n + 16)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*v[i - 2] - ((n - 2)*x + (n^2 - 6*n + 8)*x^2 - (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) + (3*n - 8 + (2*n^2 - 12*n + 18)*x)*v[i - 1] + ((n - 3)*x - (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
   1:    1;
   2:    4,      2;
   3:   11,     22,       3;
   4:   26,    132,      78,        4;
   5:   57,    604,     906,      228,        5;
   6:  120,   2382,    7248,     4764,      600,        6;
   7:  247,   8586,   46857,    62476,    21465,     1482,        7;
   8:  502,  29216,  264702,   624760,   441170,    87648,     3514,       8;
   9: 1013,  95680, 1365576,  5241416,  6551770,  2731152,   334880,    8104,     9;
  10: 2036, 305274, 6610464, 38952456, 78621240, 58428684, 15424416, 1221096, 18324, 10;

The row polynomials satisfy (n-1)*P(n,x) = -(2*n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (n*x+(n^2-2*n)*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) + (3*n-2+(2*n^2-4*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.

P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (n*x + (n^2 - 2*n)*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] + (3*n - 2 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(2*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (n*x + (n^2 - 2*n)*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) + (3*n - 2 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((n - 1)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-8d8e49a698c1586a
• Coefficients of one sign. Every nonzero row, n >= 1: 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-8d8e49a698c1586a
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-8d8e49a698c1586a
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-8d8e49a698c1586a

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 derivative_polynomial, shift 0 e-8d8e49a698c1586a

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-083b494dcfd2f637 root: theorem check

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

    • parent_provers root_domain e-ffc8a9fffd6e96cb root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• A257609 → D (shift +2)
• A008292 → D

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.