A008277 → θ+1 → x ↦ x/(x+1)

OEIS matches ?

• A383203: 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: 1;
   2: 1,    3;
   3: 1,    8,    10;
   4: 1,   17,    49,     37;
   5: 1,   34,   171,    284,     151;
   6: 1,   67,   528,   1452,    1658,     674;
   7: 1,  132,  1548,   6292,   11593,    9974,    3263;
   8: 1,  261,  4443,  25139,   66561,   90981,   62375,   17007;
   9: 1,  518, 12673,  96296,  344200,  665102,  717055,  406832,   94828;
  10: 1, 1031, 36202, 361050, 1676293, 4297703, 6480624, 5740244, 2769493, 562595;

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = ((n-4)*x^2+(3*n-11)*x^3+(3*n-10)*x^4+(n-3)*x^5)*P(n-2,x) - (x^2+4*x^3+6*x^4+4*x^5+x^6)*d/dx P(n-2,x) + (1-(n-7)*x-(3*n-13)*x^2-(3*n-9)*x^3-(n-2)*x^4)*P(n-1,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.

P[1, x_] := 1;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(((n - 4)*x^2 + (3*n - 11)*x^3 + (3*n - 10)*x^4 + (n - 3)*x^5)*P[n - 2, x] - (x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*D[P[n - 2, x], x] + (1 - (n - 7)*x - (3*n - 13)*x^2 - (3*n - 9)*x^3 - (n - 2)*x^4)*P[n - 1, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = (((n - 4)*x^2 + (3*n - 11)*x^3 + (3*n - 10)*x^4 + (n - 3)*x^5)*v[i - 2] - (x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*deriv(v[i - 2], x) + (1 - (n - 7)*x - (3*n - 13)*x^2 - (3*n - 9)*x^3 - (n - 2)*x^4)*v[i - 1] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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 [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-888961e617a41267
• 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-888961e617a41267
• 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-888961e617a41267

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 mobius, shift 0 e-888961e617a41267

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image euler_polynomial, shift 0 e-907e3c6c6275331d

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

      • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-995393ce4e49a8d6 root: theorem check

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

        • basis_map A008277 e-c52960bc5e97caa0

          Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers periodic_product e-6e85c0971e84e272 root: theorem check

              Parent-row provers of polytool polytool ?

            • preserver_image mobius, shift 0 e-ad67990681a86223

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)
        • preserver_image reversal, shift 0 e-d527982dc7c0aa3b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-8e580efc86a38a12 root: theorem check

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

            • preserver_image reversal, shift 0 e-3256729839b26765

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-995393ce4e49a8d6 (expanded above)
                • ↵ basis_map A008277 e-c52960bc5e97caa0 (expanded above)

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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.