A143495 → reverse

OEIS matches ?

• A005494: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A143495: reversed triangle, 50 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 3 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,  7,    9;
   4: 1, 12,   37,    27;
   5: 1, 18,   97,   175,     81;
   6: 1, 25,  205,   660,    781,    243;
   7: 1, 33,  380,  1890,   4081,   3367,    729;
   8: 1, 42,  644,  4550,  15421,  23772,  14197,   2187;
   9: 1, 52, 1022,  9702,  47271, 116298, 133057,  58975,   6561;
  10: 1, 63, 1542, 18900, 124887, 447195, 830845, 724260, 242461, 19683;

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

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

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n + 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); 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)

Triangle begins:
   3:     1;
   4:     3,      1;
   5:     9,      7,      1;
   6:    27,     37,     12,      1;
   7:    81,    175,     97,     18,      1;
   8:   243,    781,    660,    205,     25,      1;
   9:   729,   3367,   4081,   1890,    380,     33,     1;
  10:  2187,  14197,  23772,  15421,   4550,    644,    42,    1;
  11:  6561,  58975, 133057, 116298,  47271,   9702,  1022,   52,  1;
  12: 19683, 242461, 724260, 830845, 447195, 124887, 18900, 1542, 63, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 4, with P(3,x) = 1.

P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 2]], {n, 3, 12}] // Flatten

P(N) = {my(v = vector(N - 2)); v[1] = 1; for(n = 4, N, my(i = n - 2); v[i] = (1 + (n - 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};
T(12)

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-0c475173369de966
• 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-0c475173369de966
• 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-0c475173369de966

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 reversal, shift 0 e-0c475173369de966

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: 3-Stirling numbers of the second kind.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-4d8fac6359540276 root: theorem check

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

    • preserver_image derivative_polynomial, shift 0 e-95a6a1104a7cd304

      Laguerre’s theorem on differential operators [SP14] polytool ?

      • premise (sequence): Triangle read by rows: 2-Stirling numbers of the second kind.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-d198e25506383784 root: theorem check

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

        • preserver_image derivative_polynomial, shift 0 e-34c9c7195bdd20aa

          Laguerre’s theorem on differential operators [SP14] 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 ?

Same fingerprint (3)

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

• A038207 → basis A008277
• A120910 → basis A008277
• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.

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.