A001263 → (θ+1)²

OEIS matches ?

• A141222: row sums, 15 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:
  0: 1;
  1: 1,   4;
  2: 1,  12,    9;
  3: 1,  24,   54,    16;
  4: 1,  40,  180,   160,     25;
  5: 1,  60,  450,   800,    375,     36;
  6: 1,  84,  945,  2800,   2625,    756,     49;
  7: 1, 112, 1764,  7840,  12250,   7056,   1372,    64;
  8: 1, 144, 3024, 18816,  44100,  42336,  16464,  2304,   81;
  9: 1, 180, 4860, 40320, 132300, 190512, 123480, 34560, 3645, 100;

The row polynomials satisfy -(n^2+n-(n^4+2*n^3+n^2)*x)*P(n,x) = -(n^2+n-(n^4+2*n^3-5*n^2-2*n)*x-(3*n^4+8*n^3+4*n^2+n)*x^2)*P(n-1,x) - ((2*n+1)*x-(2*n^3+6*n^2+6*n+2)*x^2+(2*n^3+6*n^2+4*n+1)*x^3)*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 - (n^4 + 2*n^3 - 5*n^2 - 2*n)*x - (3*n^4 + 8*n^3 + 4*n^2 + n)*x^2)*P[n - 1, x] - ((2*n + 1)*x - (2*n^3 + 6*n^2 + 6*n + 2)*x^2 + (2*n^3 + 6*n^2 + 4*n + 1)*x^3)*D[P[n - 1, x], x])/(-n^2 - n + (n^4 + 2*n^3 + n^2)*x)]];
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 - (n^4 + 2*n^3 - 5*n^2 - 2*n)*x - (3*n^4 + 8*n^3 + 4*n^2 + n)*x^2)*v[i - 1] - ((2*n + 1)*x - (2*n^3 + 6*n^2 + 6*n + 2)*x^2 + (2*n^3 + 6*n^2 + 4*n + 1)*x^3)*deriv(v[i - 1], x))/(-n^2 - n + (n^4 + 2*n^3 + n^2)*x)); 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-f46dab037b613258
• 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-f46dab037b613258
• 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-f46dab037b613258
• 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-f46dab037b613258

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-f46dab037b613258

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

  • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-d285019b29e04549 root: theorem check

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

    • preserver_image reversal, shift 0 e-14da04f2a2dce119

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-d285019b29e04549 (expanded above)

Related sequences ?

Same fingerprint (2)

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

• narayana → (θ+1)²
• A367270 → θ+1

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