A136124 → Pₙ + Pₙ₊₁

OEIS matches ?

• A171005: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 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:
  0:        3,        1;
  1:        8,        6,        1;
  2:       30,       31,       10,        1;
  3:      144,      180,       80,       15,        1;
  4:      840,     1198,      651,      169,       21,       1;
  5:     5760,     9072,     5684,     1820,      315,      28,      1;
  6:    45360,    77292,    53964,    20089,     4320,     538,     36,     1;
  7:   403200,   732960,   557864,   233100,    58674,    9135,    861,    45,    1;
  8:  3991680,  7663536,  6262740,  2870440,   815815,  149513,  17710,  1310,   55,  1;
  9: 43545600, 87626880, 76044672, 37622904, 11789756, 2451834, 343497, 32076, 1914, 66, 1;

The row polynomials satisfy (n+2+x)*P(n,x) = (n^2+4*n+3+(2*n+4)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 3 + (2*n + 4)*x + x^2)*v[i - 1])/(n + 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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-3ae7b36147808534

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

• obreschkoff_combination e-3ae7b36147808534

  Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

  • premise (sequence): Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the fir…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-ab2b07f2d97ec943 root: theorem check

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

    • basis_map A007318 e-ed30910ee779b0d9

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-5f21311bb67f8c1c root: theorem check

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

        • basis_map A007318 e-0b3ba9047fae59ed

          Geometric basis map (elementary) polytool ?

          • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

              Parent-row provers of polytool polytool ?

        • parent_provers root_domain e-2e5c10abc86c264d root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-90e69a839bdbf74b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-419a6747cb084155 root: theorem check

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

            • preserver_image reversal, shift 0 e-986a227a8a459030

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-5f21311bb67f8c1c (expanded above)
                • ↵ basis_map A007318 e-0b3ba9047fae59ed (expanded above)
                • ↵ parent_provers root_domain e-2e5c10abc86c264d (expanded above)
                • preserver_image affine, shift 0 e-1ba7a8eebd893648

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ parent_provers root_domain e-edbf92ee2e446027 (expanded above)
            • parent_provers paper_theorem e-b29f35705824bef6 root: theorem check

              Parent-row provers of polytool polytool ?

        • ↵ preserver_image affine, shift 0 e-1ba7a8eebd893648 (expanded above)
    • parent_provers root_domain e-f17aff623964d25a root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image affine, shift 0 e-ea4f969963775c13

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-5f21311bb67f8c1c (expanded above)
        • ↵ basis_map A007318 e-0b3ba9047fae59ed (expanded above)
        • ↵ parent_provers root_domain e-2e5c10abc86c264d (expanded above)
        • ↵ preserver_image reversal, shift 0 e-90e69a839bdbf74b (expanded above)
        • ↵ preserver_image affine, shift 0 e-1ba7a8eebd893648 (expanded above)
  • premise (sequence): Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the fir…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 0 e-ea4f969963775c13 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
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. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.