A099093 → θ+1

OEIS matches ?

• A099432: 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: 0, 6;
  2: 0, 6, 27;
  3: 0, 0, 54, 108;
  4: 0, 0, 27, 324,  405;
  5: 0, 0,  0, 324, 1620,  1458;
  6: 0, 0,  0, 108, 2430,  7290,   5103;
  7: 0, 0,  0,   0, 1620, 14580,  30618,  17496;
  8: 0, 0,  0,   0,  405, 14580,  76545, 122472,  59049;
  9: 0, 0,  0,   0,    0,  7290, 102060, 367416, 472392, 196830;

The row polynomials satisfy n*P(n,x) = -(2*n-2-6*x)*P(n-1,x) + (4*x+3*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 - 6*x)*P[n - 1, x] + (4*x + 3*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 - 6*x)*v[i - 1] + (4*x + 3*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-cc51b36262abad1a
• 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-cc51b36262abad1a
• 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-cc51b36262abad1a

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-cc51b36262abad1a

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

  • premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A099093 e-97db6227df9d6f92

      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-a3711e6f8037cbc8 root: theorem check

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

    • basis_map A027466 e-04516916870a85c6

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
        • preserver_image affine, shift 0 e-bc143c20b95292bc

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A038243 e-089b35b0d9fc7c83

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
        • preserver_image affine, shift 0 e-ae28d4484158822f

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A007318 e-344ac1c74a22d15f

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
        • preserver_image affine, shift 0 e-9ac4433d08f91ae9

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A038231 e-3547903a43c489fd

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
        • preserver_image affine, shift 0 e-4e1348328210bdec

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A027467 e-35fdacf74fcfdf8c

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
        • preserver_image affine, shift 0 e-b320272c2500474a

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A027465 e-69bbc7837aabb73c

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
        • preserver_image affine, shift 0 e-7438d71d24adbb8f

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A038291 e-70271c52b20af48c

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
        • preserver_image affine, shift 0 e-7d49a73cf4bf68ad

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A038207 e-726dbe4e92e67f96

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
        • preserver_image affine, shift 0 e-8e25fd3c0a02e03b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A038279 e-78bb889c5d4c7dc1

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
        • preserver_image affine, shift 0 e-b4aa148b93b9dcb2

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A038303 e-a4925a412f743bbb

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
        • preserver_image affine, shift 0 e-d18687645e4e9661

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A038327 e-e0b7b31567b585e1

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
        • preserver_image affine, shift 0 e-2a73ae63953e9ff6

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
    • basis_map A038255 e-e9cb24f9617f2dae

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
        • preserver_image affine, shift 0 e-6c0da882dbe0cffe

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array (1, 3+3x).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
            • ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (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.