delannoy → D+1

OEIS matches ?

• A026937: 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:  2,   1;
  2:  4,   5,    1;
  3:  6,  15,    8,    1;
  4:  8,  33,   34,   11,    1;
  5: 10,  59,  100,   61,   14,    1;
  6: 12,  93,  230,  227,   96,   17,    1;
  7: 14, 135,  448,  645,  434,  139,   20,   1;
  8: 16, 185,  778, 1515, 1476,  741,  190,  23,  1;
  9: 18, 243, 1244, 3101, 4086, 2943, 1168, 249, 26, 1;

The row polynomials satisfy -(2*n^3-4*n^2+2*n-(2*n^3-5*n^2+12*n)*x-(4*n^2-9*n)*x^2-(n^2+2*n)*x^3-n*x^4)*P(n,x) = -(2*n^3-2*n^2+(6*n^3-15*n^2-9*n)*x+(10*n^2-32*n+23)*x^2+(3*n^2+3*n-7)*x^3+(2*n-7)*x^4-x^5)*P(n-1,x) + (2*n^2*x+(12*n^2+5*n-2)*x^2+(2*n^2+31*n-7)*x^3+(11*n+29)*x^4+(n+11)*x^5+x^6)*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^3 - 2*n^2 + (6*n^3 - 15*n^2 - 9*n)*x + (10*n^2 - 32*n + 23)*x^2 + (3*n^2 + 3*n - 7)*x^3 + (2*n - 7)*x^4 - x^5)*P[n - 1, x] + (2*n^2*x + (12*n^2 + 5*n - 2)*x^2 + (2*n^2 + 31*n - 7)*x^3 + (11*n + 29)*x^4 + (n + 11)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^3 + 4*n^2 - 2*n + (2*n^3 - 5*n^2 + 12*n)*x + (4*n^2 - 9*n)*x^2 + (n^2 + 2*n)*x^3 + n*x^4)]];
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^3 - 2*n^2 + (6*n^3 - 15*n^2 - 9*n)*x + (10*n^2 - 32*n + 23)*x^2 + (3*n^2 + 3*n - 7)*x^3 + (2*n - 7)*x^4 - x^5)*v[i - 1] + (2*n^2*x + (12*n^2 + 5*n - 2)*x^2 + (2*n^2 + 31*n - 7)*x^3 + (11*n + 29)*x^4 + (n + 11)*x^5 + x^6)*deriv(v[i - 1], x))/(-2*n^3 + 4*n^2 - 2*n + (2*n^3 - 5*n^2 + 12*n)*x + (4*n^2 - 9*n)*x^2 + (n^2 + 2*n)*x^3 + n*x^4)); 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-8cb8c25acce6a0b5
• 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-8cb8c25acce6a0b5
• 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-8cb8c25acce6a0b5
• 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-8cb8c25acce6a0b5

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-8cb8c25acce6a0b5

  Laguerre’s theorem on differential operators [SP14] 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 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 ?

Same fingerprint (1)

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

• A008288 → 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.