A011973 → ⊙ C(n,k) → θ+1

OEIS matches ?

• A391990: 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 ?

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: 1;
  1: 1;
  2: 1,   4;
  3: 1,  12;
  4: 1,  24,   18;
  5: 1,  40,   90;
  6: 1,  60,  270,   80;
  7: 1,  84,  630,  560;
  8: 1, 112, 1260, 2240,  350;
  9: 1, 144, 2268, 6720, 3150;

The row polynomials satisfy -(n-(n^3+n^2)*x)*P(n,x) = -(n-(n^3+n^2-8*n+8)*x-(4*n^3+8*n^2-4*n-8)*x^2)*P(n-1,x) - (2*x-(2*n^2+6*n+12)*x^2+(8*n^2+24*n+16)*x^3)*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 - (n^3 + n^2 - 8*n + 8)*x - (4*n^3 + 8*n^2 - 4*n - 8)*x^2)*P[n - 1, x] - (2*x - (2*n^2 + 6*n + 12)*x^2 + (8*n^2 + 24*n + 16)*x^3)*D[P[n - 1, x], x])/(-n + (n^3 + n^2)*x)]];
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 - (n^3 + n^2 - 8*n + 8)*x - (4*n^3 + 8*n^2 - 4*n - 8)*x^2)*v[i - 1] - (2*x - (2*n^2 + 6*n + 12)*x^2 + (8*n^2 + 24*n + 16)*x^3)*deriv(v[i - 1], x))/(-n + (n^3 + 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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-1cedba46f4fbf2b0, e-23c7b83afd661938
• 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-1cedba46f4fbf2b0, e-23c7b83afd661938
• 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-1cedba46f4fbf2b0, e-23c7b83afd661938

Proof chains

2 incoming proof steps. 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-1cedba46f4fbf2b0

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

  • premise (sequence): Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image euler_polynomial, shift 0 e-14cbfe8df545ebf2

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

      • premise (sequence): Triangular array of Motzkin polynomial coefficients.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • gamma_polynomial scale 1, shift 1 e-3261655b8c5a1522

          Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

          • premise (sequence): narayana
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check

              Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

            • preserver_image reversal, shift 0 e-29ef477df4172e22

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): narayana
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)
        • gamma_polynomial scale 1, shift 0 e-55e5f4941dd58e15

          Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

          • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check three_term_recurrence e-d285019b29e04549 root: theorem check

              Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

            • preserver_image reversal, shift 0 e-14da04f2a2dce119

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check three_term_recurrence e-d285019b29e04549 (expanded above)
        • preserver_image hadamard_multiplier, shift 0 e-e05d1c749616842f

          Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] 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)
    • preserver_image hadamard_multiplier, shift 0 e-2126fae20f2e5116

      Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] 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 (expanded above)
        • ↵ parent_provers paper_theorem e-0d3631d0f359c9ce (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25 (expanded above)
    • gamma_polynomial scale 1, shift 0 e-e4032a9809c053c3

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Square the entries of Pascal's triangle.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-c6b04c4560617585 root: theorem check

          Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

        • preserver_image reversal, shift 0 e-1906182487ade7a7

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Square the entries of Pascal's triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-c6b04c4560617585 (expanded above)
• preserver_image euler_polynomial, shift 0 e-23c7b83afd661938

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

  • premise (sequence): Triangular array of Motzkin polynomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ gamma_polynomial scale 1, shift 1 e-3261655b8c5a1522 (expanded above)
    • ↵ gamma_polynomial scale 1, shift 0 e-55e5f4941dd58e15 (expanded above)
    • ↵ preserver_image hadamard_multiplier, shift 0 e-e05d1c749616842f (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A059344 → ⊙ 1/k! (shift -1) → 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023