A172431 → ⊙ C(n,k+1) (shift -1) → (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:
   1: 1;
   2: 1,   4;
   3: 1,  16,     9;
   4: 1,  36,    90,     16;
   5: 1,  64,   378,    320,     25;
   6: 1, 100,  1080,   2240,    875,      36;
   7: 1, 144,  2475,   9600,   9450,    2016,      49;
   8: 1, 196,  4914,  30800,  57750,   31752,    4116,     64;
   9: 1, 256,  8820,  81536, 250250,  266112,   90552,   7680,    81;
  10: 1, 324, 14688, 188160, 859950, 1513512, 1009008, 228096, 13365, 100;

The row polynomials satisfy -(8*n^4-24*n^3+26*n^2-12*n+2-(16*n^6-80*n^5+164*n^4-176*n^3+104*n^2-32*n+4)*x)*P(n,x) = -(8*n^4-24*n^3+26*n^2-12*n+2-(16*n^6-80*n^5+20*n^4+424*n^3-768*n^2+480*n-100)*x-(144*n^6-664*n^5+1000*n^4-316*n^3-548*n^2+512*n-128)*x^2)*P(n-1,x) - ((16*n^3-44*n^2+34*n-8)*x-(32*n^5-80*n^4+148*n^3-324*n^2+258*n-62)*x^2+(208*n^5-704*n^4+500*n^3+380*n^2-544*n+160)*x^3)*d/dx P(n-1,x) + ((10*n^2-11*n+3)*x^2-(20*n^4-26*n^3+30*n^2-20*n+4)*x^3+(80*n^4-104*n^3-40*n^2+96*n-32)*x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^4 - 24*n^3 + 26*n^2 - 12*n + 2 - (16*n^6 - 80*n^5 + 20*n^4 + 424*n^3 - 768*n^2 + 480*n - 100)*x - (144*n^6 - 664*n^5 + 1000*n^4 - 316*n^3 - 548*n^2 + 512*n - 128)*x^2)*P[n - 1, x] - ((16*n^3 - 44*n^2 + 34*n - 8)*x - (32*n^5 - 80*n^4 + 148*n^3 - 324*n^2 + 258*n - 62)*x^2 + (208*n^5 - 704*n^4 + 500*n^3 + 380*n^2 - 544*n + 160)*x^3)*D[P[n - 1, x], x] + ((10*n^2 - 11*n + 3)*x^2 - (20*n^4 - 26*n^3 + 30*n^2 - 20*n + 4)*x^3 + (80*n^4 - 104*n^3 - 40*n^2 + 96*n - 32)*x^4)*D[P[n - 1, x], {x, 2}])/(-8*n^4 + 24*n^3 - 26*n^2 + 12*n - 2 + (16*n^6 - 80*n^5 + 164*n^4 - 176*n^3 + 104*n^2 - 32*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(8*n^4 - 24*n^3 + 26*n^2 - 12*n + 2 - (16*n^6 - 80*n^5 + 20*n^4 + 424*n^3 - 768*n^2 + 480*n - 100)*x - (144*n^6 - 664*n^5 + 1000*n^4 - 316*n^3 - 548*n^2 + 512*n - 128)*x^2)*v[i - 1] - ((16*n^3 - 44*n^2 + 34*n - 8)*x - (32*n^5 - 80*n^4 + 148*n^3 - 324*n^2 + 258*n - 62)*x^2 + (208*n^5 - 704*n^4 + 500*n^3 + 380*n^2 - 544*n + 160)*x^3)*deriv(v[i - 1], x) + ((10*n^2 - 11*n + 3)*x^2 - (20*n^4 - 26*n^3 + 30*n^2 - 20*n + 4)*x^3 + (80*n^4 - 104*n^3 - 40*n^2 + 96*n - 32)*x^4)*deriv(deriv(v[i - 1], x), x))/(-8*n^4 + 24*n^3 - 26*n^2 + 12*n - 2 + (16*n^6 - 80*n^5 + 164*n^4 - 176*n^3 + 104*n^2 - 32*n + 4)*x)); 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-e9c68bdfe2122a78
• 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-e9c68bdfe2122a78
• 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-e9c68bdfe2122a78

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-e9c68bdfe2122a78

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

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image hadamard_multiplier, shift -1 e-1491609a6fc5f8c9

      Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

      • premise (sequence): Even row Pascal-square read by antidiagonals.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-83282836b5ac538e 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