A008278 → x ↦ x+1 (shift +1) → x ↦ x/(x+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:      2,       3;
  2:      5,      15,       11;
  3:     15,      68,      101,       49;
  4:     52,     317,      715,      706,      257;
  5:    203,    1559,     4737,     7111,     5268,      1539;
  6:    877,    8138,    31176,    63068,    71004,     42149,     10299;
  7:   4140,   45093,   208826,   532724,   808067,    728411,    361099,    75905;
  8:  21147,  264671,  1439145,  4438642,  8489401,  10306162,   7752138,  3301796,   609441;
  9: 115975, 1640930, 10254791, 37139185, 85873202, 131415497, 133059970, 85924298, 32100422, 5284451;

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2*n+(10*n-3)*x+(18*n-10)*x^2+(14*n-11)*x^3+(4*n-4)*x^4)*P(n-1,x) - (1+7*x+19*x^2+25*x^3+16*x^4+4*x^5)*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[((2*n + (10*n - 3)*x + (18*n - 10)*x^2 + (14*n - 11)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 4*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = ((2*n + (10*n - 3)*x + (18*n - 10)*x^2 + (14*n - 11)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 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 - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, 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-2aa326c0b7668d12
• 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-2aa326c0b7668d12
• 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-2aa326c0b7668d12

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-2aa326c0b7668d12

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image affine, shift 1 e-236d0fb15dc6eca0

      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 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 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [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
2. [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