A257609 → D (shift +2)

OEIS matches ?

• A001286: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A180119: 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:
   1:   11,      22,         3;
   2:   26,     132,        78,          4;
   3:   57,     604,       906,        228,          5;
   4:  120,    2382,      7248,       4764,        600,           6;
   5:  247,    8586,     46857,      62476,      21465,        1482,           7;
   6:  502,   29216,    264702,     624760,     441170,       87648,        3514,          8;
   7: 1013,   95680,   1365576,    5241416,    6551770,     2731152,      334880,       8104,         9;
   8: 2036,  305274,   6610464,   38952456,   78621240,    58428684,    15424416,    1221096,     18324,       10;
   9: 4083,  956542,  30563055,  265273896,  812561430,   975073716,   464229318,   81501480,   4304439,    40830,    11;
  10: 8178, 2959452, 136600350, 1693126140, 7528107540, 13651032024, 10539350556, 3386252280, 409801050, 14797260, 89958, 12;

The row polynomials satisfy (n+1)*P(n,x) = (3*n+4+(2*n^2+4*n+2)*x)*P(n-1,x) + ((n+1)*x-(n+1)*x^2)*d/dx P(n-1,x) - (2*n+4+(2*n^2+4*n)*x+(n^3+3*n^2+2*n)*x^2)*P(n-2,x) - ((n+2)*x+(n^2+2*n)*x^2-(n^2+3*n+2)*x^3)*d/dx P(n-2,x) for n >= 3, with P(1,x) = 11 + 22*x + 3*x^2, P(2,x) = 26 + 132*x + 78*x^2 + 4*x^3.

P[1, x_] := 11 + 22*x + 3*x^2;
P[2, x_] := 26 + 132*x + 78*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 4 + (2*n^2 + 4*n + 2)*x)*P[n - 1, x] + ((n + 1)*x - (n + 1)*x^2)*D[P[n - 1, x], x] - (2*n + 4 + (2*n^2 + 4*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*P[n - 2, x] - ((n + 2)*x + (n^2 + 2*n)*x^2 - (n^2 + 3*n + 2)*x^3)*D[P[n - 2, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 11 + 22*x + 3*x^2; v[2] = 26 + 132*x + 78*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = ((3*n + 4 + (2*n^2 + 4*n + 2)*x)*v[i - 1] + ((n + 1)*x - (n + 1)*x^2)*deriv(v[i - 1], x) - (2*n + 4 + (2*n^2 + 4*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*v[i - 2] - ((n + 2)*x + (n^2 + 2*n)*x^2 - (n^2 + 3*n + 2)*x^3)*deriv(v[i - 2], x))/(n + 1)); 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)

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-2ca40a069e1a39d6
• 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-2ca40a069e1a39d6
• 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-2ca40a069e1a39d6
• 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-2ca40a069e1a39d6

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 2 e-2ca40a069e1a39d6

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

  • premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.
    uses: Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-7ce0c29875c331b6 root: theorem check

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

    • parent_provers root_domain e-f2db3ed118a27f94 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.

• A173018 → D
• 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.