A145596 → D+1

OEIS matches ?

• A002694: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 2 ?

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:   4,    2;
  2:  11,   14,     3;
  3:  24,   60,    32,      4;
  4:  45,  190,   195,     60,      5;
  5:  76,  490,   840,    490,    100,      6;
  6: 119, 1092,  2842,   2744,   1050,    154,     7;
  7: 176, 2184,  8064,  11760,   7392,   2016,   224,    8;
  8: 249, 4020, 20034,  41328,  39060,  17388,  3570,  312,   9;
  9: 340, 6930, 44880, 124740, 166320, 110880, 36960, 5940, 420, 10;

The row polynomials satisfy (n^4+4*n^3+2*n^2+8*n+(2*n^3+6*n^2-8*n)*x+(n^2+4*n)*x^2)*P(n,x) = (n^4+7*n^3+17*n^2+23*n+12+(3*n^4+9*n^3+16*n^2+6*n-4)*x+(6*n^3+8*n^2+4*n+2)*x^2+(3*n^2+5*n+2)*x^3)*P(n-1,x) + ((2*n^3+5*n^2+7*n+4)*x-(2*n^3+n^2+3*n+4)*x^2-(4*n^2+2*n-2)*x^3-(2*n+2)*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[((n^4 + 7*n^3 + 17*n^2 + 23*n + 12 + (3*n^4 + 9*n^3 + 16*n^2 + 6*n - 4)*x + (6*n^3 + 8*n^2 + 4*n + 2)*x^2 + (3*n^2 + 5*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 + 5*n^2 + 7*n + 4)*x - (2*n^3 + n^2 + 3*n + 4)*x^2 - (4*n^2 + 2*n - 2)*x^3 - (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 4*n^3 + 2*n^2 + 8*n + (2*n^3 + 6*n^2 - 8*n)*x + (n^2 + 4*n)*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] = ((n^4 + 7*n^3 + 17*n^2 + 23*n + 12 + (3*n^4 + 9*n^3 + 16*n^2 + 6*n - 4)*x + (6*n^3 + 8*n^2 + 4*n + 2)*x^2 + (3*n^2 + 5*n + 2)*x^3)*v[i - 1] + ((2*n^3 + 5*n^2 + 7*n + 4)*x - (2*n^3 + n^2 + 3*n + 4)*x^2 - (4*n^2 + 2*n - 2)*x^3 - (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 4*n^3 + 2*n^2 + 8*n + (2*n^3 + 6*n^2 - 8*n)*x + (n^2 + 4*n)*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-92e33d8ed5abb5e6
• 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-92e33d8ed5abb5e6
• 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-92e33d8ed5abb5e6
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-92e33d8ed5abb5e6

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-92e33d8ed5abb5e6

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

  • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c483eeee9c2638c9 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • preserver_image reversal, shift 0 e-1d2c7b4a3cb3d72f

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-c483eeee9c2638c9 (expanded above)
        • basis_map A007318 e-fb63394f9047a486

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
            • preserver_image affine, shift 0 e-3babd9179153850c

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-c483eeee9c2638c9 (expanded above)
    • ↵ basis_map A007318 e-fb63394f9047a486 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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