A011973 → x ↦ x+2 (shift +1) → θ+1

OEIS matches ?

No OEIS matches.

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:   3,    2;
  2:   5,    4;
  3:  11,   14,    3;
  4:  21,   32,    9;
  5:  43,   82,   36,   4;
  6:  85,  188,  102,  16;
  7: 171,  438,  297,  72,   5;
  8: 341,  984,  783, 240,  25;
  9: 683, 2202, 2034, 780, 125, 6;

The row polynomials satisfy (36*n+36+(9*n^2+31*n+22)*x+(n^3+4*n^2+5*n+2)*x^2)*P(n,x) = (180*n+36+(63*n^2+245*n+22)*x+(5*n^3+52*n^2+101*n+2)*x^2+(2*n^3+10*n^2+12*n)*x^3)*P(n-1,x) - (324+(162*n+702)*x+(18*n^2+243*n+554)*x^2+(17*n^2+121*n+190)*x^3+(4*n^2+20*n+24)*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[((180*n + 36 + (63*n^2 + 245*n + 22)*x + (5*n^3 + 52*n^2 + 101*n + 2)*x^2 + (2*n^3 + 10*n^2 + 12*n)*x^3)*P[n - 1, x] - (324 + (162*n + 702)*x + (18*n^2 + 243*n + 554)*x^2 + (17*n^2 + 121*n + 190)*x^3 + (4*n^2 + 20*n + 24)*x^4)*D[P[n - 1, x], x])/(36*n + 36 + (9*n^2 + 31*n + 22)*x + (n^3 + 4*n^2 + 5*n + 2)*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] = ((180*n + 36 + (63*n^2 + 245*n + 22)*x + (5*n^3 + 52*n^2 + 101*n + 2)*x^2 + (2*n^3 + 10*n^2 + 12*n)*x^3)*v[i - 1] - (324 + (162*n + 702)*x + (18*n^2 + 243*n + 554)*x^2 + (17*n^2 + 121*n + 190)*x^3 + (4*n^2 + 20*n + 24)*x^4)*deriv(v[i - 1], x))/(36*n + 36 + (9*n^2 + 31*n + 22)*x + (n^3 + 4*n^2 + 5*n + 2)*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-42735f4be786e5b0
• 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-42735f4be786e5b0
• 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-42735f4be786e5b0

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 euler_polynomial, shift 0 e-42735f4be786e5b0

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

  • premise (sequence): Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image affine, shift 1 e-7fc1b55fb0aa373a

      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)

Related sequences ?

No candidate equalities or related sequences.

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