A008278 → θ+1

OEIS matches ?

• A359109: 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,  2;
   3: 1,  6,    3;
   4: 1, 12,   21,     4;
   5: 1, 20,   75,    60,      5;
   6: 1, 30,  195,   360,    155,      6;
   7: 1, 42,  420,  1400,   1505,    378,      7;
   8: 1, 56,  798,  4200,   8505,   5796,    889,     8;
   9: 1, 72, 1386, 10584,  34755,  46620,  21175,  2040,    9;
  10: 1, 90, 2250, 23520, 114135, 255150, 238735, 74640, 4599, 10;

The row polynomials satisfy (n-1)*P(n,x) = -(n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (n*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n-1+(2*n^2-4*n+2)*x)*P(n-1,x) - (n-1)*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.

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

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (n*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n - 1 + (2*n^2 - 4*n + 2)*x)*v[i - 1] - (n - 1)*x^2*deriv(v[i - 1], x))/(n - 1)); 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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-162681880755ce3c
• 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-162681880755ce3c
• 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-162681880755ce3c
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-162681880755ce3c

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-162681880755ce3c

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

Same fingerprint (1)

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

• A106800 → 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. [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.