A030528 → D

OEIS matches ?

• A004798: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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: 0;
  1: 1;
  2: 2, 2;
  3: 1, 6,  3;
  4: 0, 6, 12,  4;
  5: 0, 2, 18, 20,   5;
  6: 0, 0, 12, 40,  30,   6;
  7: 0, 0,  3, 40,  75,  42,   7;
  8: 0, 0,  0, 20, 100, 126,  56,  8;
  9: 0, 0,  0,  4,  75, 210, 196, 72, 9;

The row polynomials satisfy (n^3-2*n^2-n+2-(2*n^2-2)*x)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(8*n^2-24*n+4)*x+6*n*x^2)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2-11*n+2)*x^2-2*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n^2 - 24*n + 4)*x + 6*n*x^2)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 - 11*n + 2)*x^2 - 2*n*x^3)*D[P[n - 1, x], x])/(n^3 - 2*n^2 - n + 2 - (2*n^2 - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n^2 - 24*n + 4)*x + 6*n*x^2)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 - 11*n + 2)*x^2 - 2*n*x^3)*deriv(v[i - 1], x))/(n^3 - 2*n^2 - n + 2 - (2*n^2 - 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-b66135a965ab5be8
• 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-b66135a965ab5be8
• 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-b66135a965ab5be8

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-b66135a965ab5be8

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

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A027466 e-1781530cbfe1e9bf

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
        • preserver_image affine, shift 0 e-742e5241e9a4e627

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
            • theorem_check ma_wang_interval e-f811a014585c6352 root: theorem check

              Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

    • basis_map A027467 e-361937c2693b166b

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
        • preserver_image affine, shift 0 e-8f58f94d7421b40d

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
            • ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    • basis_map A027465 e-5c2b8bc6b43c730e

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
        • preserver_image affine, shift 0 e-a2e7017ebd9e3749

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
            • ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    • basis_map A007318 e-ba7f4c6c74dd05c0

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
        • preserver_image affine, shift 0 e-51281bad32c8f5cd

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
            • ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    • preserver_image reversal, shift 1 e-fb809a8e7c094d2b

      Affine, reversal and Möbius substitutions (elementary) 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 >= 1, 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)
    • basis_map A030528 e-be49b310b519d57a

      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
    • ↵ theorem_check ma_wang_interval e-f811a014585c6352 (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.