A178820 → θ+1

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:   1;
  1:   4,    8;
  2:  10,   40,    30;
  3:  20,  120,   180,    80;
  4:  35,  280,   630,   560,    175;
  5:  56,  560,  1680,  2240,   1400,    336;
  6:  84, 1008,  3780,  6720,   6300,   3024,    588;
  7: 120, 1680,  7560, 16800,  21000,  15120,   5880,   960;
  8: 165, 2640, 13860, 36960,  57750,  55440,  32340, 10560,  1485;
  9: 220, 3960, 23760, 73920, 138600, 166320, 129360, 63360, 17820, 2200;

The row polynomials satisfy (n+n^2*x)*P(n,x) = (n+3+(n^2+5*n+6)*x+(n^2+4*n+3)*x^2)*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 + 3 + (n^2 + 5*n + 6)*x + (n^2 + 4*n + 3)*x^2)*P[n - 1, x])/(n + 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 + 3 + (n^2 + 5*n + 6)*x + (n^2 + 4*n + 3)*x^2)*v[i - 1])/(n + 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-c68d4e8c3af46596
• 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-c68d4e8c3af46596
• 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-c68d4e8c3af46596
• 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-c68d4e8c3af46596

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-c68d4e8c3af46596

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

  • premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-387009a21707feb6 root: theorem check

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

    • preserver_image reversal, shift 0 e-17b3b03bdb024482

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-387009a21707feb6 (expanded above)

Related sequences ?

Same fingerprint (27)

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

• A344565 → θ+1
• A028326 → θ+1
• A178821 → D+1
• basis-map coefficients of A113214, shifted → D+1
• A007318 → ×(1+2n+n²) → D+1
• A085880 → D+1
• A178822 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A135838 → D+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → θ+1
• A007318 → D → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A178822 → D+1
• A038208 → D+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → D+1
• A007318 → D → θ+1
• A007318 → ×(1+2n+n²) → θ+1
• basis-map coefficients of A113214, shifted → θ+1
• A028326 → D+1
• A085880 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A178820 → D+1
• A178821 → θ+1
• A344565 → D+1
• A038208 → θ+1
• A135838 → 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. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.