A052553 → 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:
   1: 1;
   2: 1;
   3: 2,  1;
   4: 3,  2;
   5: 4,  5,  1;
   6: 5, 10,  3;
   7: 6, 17,  9,  1;
   8: 7, 26, 22,  4;
   9: 8, 37, 45, 14, 1;
  10: 9, 50, 81, 40, 5;

The row polynomials satisfy (n^3-5*n^2+8*n-4+(4*n^2-17*n+13)*x+(4*n-4)*x^2)*P(n,x) = (n^3-4*n^2+5*n-2+(2*n^3-8*n^2+5*n+1)*x+(8*n^2-30*n+32)*x^2+(8*n-16)*x^3)*P(n-1,x) - ((n^2-2*n+1)*x+(4*n^2-4*n-1)*x^2+(16*n-16)*x^3+16*x^4)*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[Together[((n^3 - 4*n^2 + 5*n - 2 + (2*n^3 - 8*n^2 + 5*n + 1)*x + (8*n^2 - 30*n + 32)*x^2 + (8*n - 16)*x^3)*P[n - 1, x] - ((n^2 - 2*n + 1)*x + (4*n^2 - 4*n - 1)*x^2 + (16*n - 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - 5*n^2 + 8*n - 4 + (4*n^2 - 17*n + 13)*x + (4*n - 4)*x^2)]];
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] = ((n^3 - 4*n^2 + 5*n - 2 + (2*n^3 - 8*n^2 + 5*n + 1)*x + (8*n^2 - 30*n + 32)*x^2 + (8*n - 16)*x^3)*v[i - 1] - ((n^2 - 2*n + 1)*x + (4*n^2 - 4*n - 1)*x^2 + (16*n - 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - 5*n^2 + 8*n - 4 + (4*n^2 - 17*n + 13)*x + (4*n - 4)*x^2)); 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)

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-0011a8ca306f5399
• 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-0011a8ca306f5399
• 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-0011a8ca306f5399

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-0011a8ca306f5399

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

  • premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1, shift 0 e-01e2cd957653d3da

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang_derivative_lag e-b176436450506e78 root: theorem check

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

        • preserver_image reversal, shift 0 e-73f1cbc14f4fad08

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang_derivative_lag e-b176436450506e78 (expanded above)
            • basis_map A008288 e-796ce5140a0b4b04

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_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 [0, 0]
                no recorded property covers this requirement
            • basis_map A008288 e-b8e0a0fa3ba7bf57

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_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 [0, 0]
                no recorded property covers this requirement
        • ↵ basis_map A008288 e-796ce5140a0b4b04 (expanded above)
        • ↵ basis_map A008288 e-b8e0a0fa3ba7bf57 (expanded above)
    • preserver_image reversal, shift 0 e-515982525d76b9fb

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • preserver_image reversal, shift -1 e-24564bef34b2c2e5

          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 >= 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)
        • basis_map A007318 e-fcd19baf8f835cf0

          Geometric basis map (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, 1]
            • preserver_image affine, shift 0 e-1365676e8cb0fd13

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
                • theorem_check ma_wang_interval e-32a5b77539bbfdf5 root: theorem check

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

            • preserver_image affine, shift 0 e-3c4e27dedb9c54a8

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0]
                • theorem_check ma_wang_interval e-0178a3d1cd265fb6 root: theorem check

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

        • basis_map A026729 e-fa68694d112454ed

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_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 [0, 0]
            no recorded property covers this requirement
        • ↵ theorem_check ma_wang_interval e-0178a3d1cd265fb6 (expanded above)
    • parent_provers paper_theorem e-cf6c2292b090b00e root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• A169803 → D+1
• A011973 → 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. [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.