A099040 → θ+1

OEIS matches ?

• A073388: 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 ?
• A109634: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 0, 4;
  2: 0, 4, 12;
  3: 0, 0, 24, 32;
  4: 0, 0, 12, 96,  80;
  5: 0, 0,  0, 96, 320,  192;
  6: 0, 0,  0, 32, 480,  960,  448;
  7: 0, 0,  0,  0, 320, 1920, 2688,  1024;
  8: 0, 0,  0,  0,  80, 1920, 6720,  7168,  2304;
  9: 0, 0,  0,  0,   0,  960, 8960, 21504, 18432, 5120;

The row polynomials satisfy n*P(n,x) = -(2*n-2-4*x)*P(n-1,x) + (4*x+2*x^2)*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[(-(2*n - 2 - 4*x)*P[n - 1, x] + (4*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];
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] = (-(2*n - 2 - 4*x)*v[i - 1] + (4*x + 2*x^2)*deriv(v[i - 1], x))/(n)); 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-d427c42eda759629
• 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-d427c42eda759629
• 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-d427c42eda759629

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-d427c42eda759629

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

  • premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A099040 e-52040eb0dce100d3

      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-17786677d2ed021b root: theorem check

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

    • basis_map A027467 e-0a457ba83443136e

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
        • preserver_image affine, shift 0 e-48e4dfa7d694f880

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A038255 e-162624a94908f408

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
        • preserver_image affine, shift 0 e-79e1a4a5e521405c

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A027466 e-3f03a2606d2e6662

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
        • preserver_image affine, shift 0 e-5e00209341210844

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A038303 e-532ccda39f04a0f0

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
        • preserver_image affine, shift 0 e-c2f106499058af8f

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A038207 e-72107609b2d5e6bc

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
        • preserver_image affine, shift 0 e-8c4e4f825f4ab306

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A038243 e-763de8a298cf812d

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
        • preserver_image affine, shift 0 e-bdc3e5aca0457f81

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A038279 e-8232f658d776d13c

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
        • preserver_image affine, shift 0 e-4478346993b2c814

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A027465 e-af79ef336fd9f8da

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
        • preserver_image affine, shift 0 e-319b2a76ea6c1ab0

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A038291 e-c3d762863d6658d7

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
        • preserver_image affine, shift 0 e-b9371054cfe1d371

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A007318 e-e158703ce212fb71

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
        • preserver_image affine, shift 0 e-dc7a132d665751c2

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A038231 e-e36b44dfb8fba667

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
        • preserver_image affine, shift 0 e-23d8b9b067cbe12f

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
    • basis_map A038327 e-ffeb0648278e3cee

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
        • preserver_image affine, shift 0 e-1000187f132dd1ea

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 2+2x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
            • ↵ theorem_check ma_wang_interval e-17786677d2ed021b (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. [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.