A011973 → D+1

OEIS matches ?

• A010049: 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:
  0: 1;
  1: 1;
  2: 2,  1;
  3: 3,  2;
  4: 4,  5,  1;
  5: 5, 10,  3;
  6: 6, 17,  9,  1;
  7: 7, 26, 22,  4;
  8: 8, 37, 45, 14, 1;
  9: 9, 50, 81, 40, 5;

The row polynomials satisfy (n^3-2*n^2+n+(4*n^2-9*n)*x+4*n*x^2)*P(n,x) = (n^3-n^2+(2*n^3-2*n^2-5*n)*x+(8*n^2-14*n+10)*x^2+(8*n-8)*x^3)*P(n-1,x) - (n^2*x+(4*n^2+4*n-1)*x^2+16*n*x^3+16*x^4)*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[((n^3 - n^2 + (2*n^3 - 2*n^2 - 5*n)*x + (8*n^2 - 14*n + 10)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (n^2*x + (4*n^2 + 4*n - 1)*x^2 + 16*n*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + n + (4*n^2 - 9*n)*x + 4*n*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] = ((n^3 - n^2 + (2*n^3 - 2*n^2 - 5*n)*x + (8*n^2 - 14*n + 10)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (n^2*x + (4*n^2 + 4*n - 1)*x^2 + 16*n*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + n + (4*n^2 - 9*n)*x + 4*n*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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-797e24a252300d6f
• 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-797e24a252300d6f
• 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-797e24a252300d6f

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-797e24a252300d6f

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

  • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check

      Parent-row provers of polytool polytool ?

    • parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check

      Parent-row provers of polytool polytool ?

    • gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): delannoy
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang_derivative_lag e-15c233e4385971d1 root: theorem check

          Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

        • preserver_image reversal, shift 0 e-266d85ca03a298fa

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): delannoy
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang_derivative_lag e-15c233e4385971d1 (expanded above)
            • basis_map A008288 e-e771e35dc9ee3fbd

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
            • basis_map A008288 e-33dba7e79fe9caa8

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
        • ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
        • ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)

Related sequences ?

Same fingerprint (2)

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

• A169803 → D+1
• A052553 → D+1

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.