A008278 → x ↦ x+1 (shift +1) → D+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:      3,       1;
  2:     10,       7,       1;
  3:     38,      43,      13,       1;
  4:    161,     261,     133,      23,       1;
  5:    747,    1606,    1203,     368,      41,       1;
  6:   3753,   10158,   10253,    4728,     976,      75,      1;
  7:  20253,   66529,   85475,   53973,   17216,    2553,    141,     1;
  8: 116642,  452623,  711457,  578315,  256826,   60359,   6679,   271,   1;
  9: 713130, 3202057, 5981815, 5997680, 3468608, 1154669, 207985, 17614, 529, 1;

The row polynomials satisfy (n+1+2*x)*P(n,x) = -(n^3+2*n^2-n-2+(2*n^3+5*n^2-3*n-4)*x+(n^3+6*n^2-2*n-2)*x^2+(3*n^2+n-2)*x^3+(2*n-2)*x^4)*P(n-2,x) + (n^2+3*n+2+(3*n^2+11*n+8)*x+(3*n^2+16*n+14)*x^2+(n^2+11*n+14)*x^3+(3*n+8)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2+3*n+1+(2*n^2+6*n)*x+5*n*x^2+2*x^3)*P(n-1,x) - (n+1+(2*n+4)*x+(n+5)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 2*n^2 - n - 2 + (2*n^3 + 5*n^2 - 3*n - 4)*x + (n^3 + 6*n^2 - 2*n - 2)*x^2 + (3*n^2 + n - 2)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + (n^2 + 3*n + 2 + (3*n^2 + 11*n + 8)*x + (3*n^2 + 16*n + 14)*x^2 + (n^2 + 11*n + 14)*x^3 + (3*n + 8)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 + 3*n + 1 + (2*n^2 + 6*n)*x + 5*n*x^2 + 2*x^3)*P[n - 1, x] - (n + 1 + (2*n + 4)*x + (n + 5)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 2*n^2 - n - 2 + (2*n^3 + 5*n^2 - 3*n - 4)*x + (n^3 + 6*n^2 - 2*n - 2)*x^2 + (3*n^2 + n - 2)*x^3 + (2*n - 2)*x^4)*v[i - 2] + (n^2 + 3*n + 2 + (3*n^2 + 11*n + 8)*x + (3*n^2 + 16*n + 14)*x^2 + (n^2 + 11*n + 14)*x^3 + (3*n + 8)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 + 3*n + 1 + (2*n^2 + 6*n)*x + 5*n*x^2 + 2*x^3)*v[i - 1] - (n + 1 + (2*n + 4)*x + (n + 5)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 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-5df37d25263414d4
• 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-5df37d25263414d4
• 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-5df37d25263414d4

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 derivative_polynomial, shift 0 e-5df37d25263414d4

  Laguerre’s theorem on differential operators [SP14] 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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.