A207606 → (θ+1)²

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:  4,   28,    18;
  1:  5,   64,    99,     32;
  2:  6,  120,   324,    240,     50;
  3:  7,  200,   819,   1024,    475,     72;
  4:  8,  308,  1764,   3264,   2500,    828,     98;
  5:  9,  448,  3402,   8640,   9625,   5184,   1323,    128;
  6: 10,  624,  6048,  20064,  30250,  23400,   9604,   1984,    162;
  7: 11,  840, 10098,  42240,  82225,  85176,  49735,  16384,   2835,   200;
  8: 12, 1100, 16038,  82368, 200200, 265356, 205800,  95744,  26244,  3900,  242;
  9: 13, 1408, 24453, 151008, 446875, 733824, 723044, 443904, 170829, 40000, 5203, 288;

The row polynomials satisfy -(8*n+24-(20*n^3+180*n^2+564*n+516)*x+(2*n^5+30*n^4+228*n^3+696*n^2+906*n+426)*x^2-(4*n^5+30*n^4+88*n^3+126*n^2+88*n+24)*x^3)*P(n,x) = -(8*n+32-(20*n^3+200*n^2+668*n+592)*x+(2*n^5+32*n^4+234*n^3+582*n^2+122*n-680)*x^2-(3*n^5+12*n^4-97*n^3-634*n^2-1200*n-736)*x^3-(2*n^5+20*n^4+76*n^3+136*n^2+114*n+36)*x^4)*P(n-1,x) - (16*x-(40*n^2+280*n+692)*x^2+(4*n^4+56*n^3+410*n^2+1066*n+722)*x^3-(7*n^4+50*n^3+71*n^2-92*n-152)*x^4-(2*n^4+16*n^3+44*n^2+48*n+18)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 28*x + 18*x^2.

P[0, x_] := 4 + 28*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 32 - (20*n^3 + 200*n^2 + 668*n + 592)*x + (2*n^5 + 32*n^4 + 234*n^3 + 582*n^2 + 122*n - 680)*x^2 - (3*n^5 + 12*n^4 - 97*n^3 - 634*n^2 - 1200*n - 736)*x^3 - (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^4)*P[n - 1, x] - (16*x - (40*n^2 + 280*n + 692)*x^2 + (4*n^4 + 56*n^3 + 410*n^2 + 1066*n + 722)*x^3 - (7*n^4 + 50*n^3 + 71*n^2 - 92*n - 152)*x^4 - (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*D[P[n - 1, x], x])/(-8*n - 24 + (20*n^3 + 180*n^2 + 564*n + 516)*x - (2*n^5 + 30*n^4 + 228*n^3 + 696*n^2 + 906*n + 426)*x^2 + (4*n^5 + 30*n^4 + 88*n^3 + 126*n^2 + 88*n + 24)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 28*x + 18*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n + 32 - (20*n^3 + 200*n^2 + 668*n + 592)*x + (2*n^5 + 32*n^4 + 234*n^3 + 582*n^2 + 122*n - 680)*x^2 - (3*n^5 + 12*n^4 - 97*n^3 - 634*n^2 - 1200*n - 736)*x^3 - (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^4)*v[i - 1] - (16*x - (40*n^2 + 280*n + 692)*x^2 + (4*n^4 + 56*n^3 + 410*n^2 + 1066*n + 722)*x^3 - (7*n^4 + 50*n^3 + 71*n^2 - 92*n - 152)*x^4 - (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*deriv(v[i - 1], x))/(-8*n - 24 + (20*n^3 + 180*n^2 + 564*n + 516)*x - (2*n^5 + 30*n^4 + 228*n^3 + 696*n^2 + 906*n + 426)*x^2 + (4*n^5 + 30*n^4 + 88*n^3 + 126*n^2 + 88*n + 24)*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-8d9c8e34f3c4da6c
• 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-8d9c8e34f3c4da6c
• 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-8d9c8e34f3c4da6c
• 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-8d9c8e34f3c4da6c

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-8d9c8e34f3c4da6c

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

  • premise (sequence): Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-62ac8b7225a515c7 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • theorem_check three_term_recurrence e-3222de37fcca30b3 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023