A008288 → 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:
   1:  1;
   2:  2,   1;
   3:  4,   5,    1;
   4:  6,  15,    8,    1;
   5:  8,  33,   34,   11,    1;
   6: 10,  59,  100,   61,   14,    1;
   7: 12,  93,  230,  227,   96,   17,    1;
   8: 14, 135,  448,  645,  434,  139,   20,   1;
   9: 16, 185,  778, 1515, 1476,  741,  190,  23,  1;
  10: 18, 243, 1244, 3101, 4086, 2943, 1168, 249, 26, 1;

The row polynomials satisfy -(2*n^3-10*n^2+16*n-8-(2*n^3-11*n^2+28*n-19)*x-(4*n^2-17*n+13)*x^2-(n^2-1)*x^3-(n-1)*x^4)*P(n,x) = -(2*n^3-8*n^2+10*n-4+(6*n^3-33*n^2+39*n-12)*x+(10*n^2-52*n+65)*x^2+(3*n^2-3*n-7)*x^3+(2*n-9)*x^4-x^5)*P(n-1,x) + ((2*n^2-4*n+2)*x+(12*n^2-19*n+5)*x^2+(2*n^2+27*n-36)*x^3+(11*n+18)*x^4+(n+10)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 8*n^2 + 10*n - 4 + (6*n^3 - 33*n^2 + 39*n - 12)*x + (10*n^2 - 52*n + 65)*x^2 + (3*n^2 - 3*n - 7)*x^3 + (2*n - 9)*x^4 - x^5)*P[n - 1, x] + ((2*n^2 - 4*n + 2)*x + (12*n^2 - 19*n + 5)*x^2 + (2*n^2 + 27*n - 36)*x^3 + (11*n + 18)*x^4 + (n + 10)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^3 + 10*n^2 - 16*n + 8 + (2*n^3 - 11*n^2 + 28*n - 19)*x + (4*n^2 - 17*n + 13)*x^2 + (n^2 - 1)*x^3 + (n - 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 8*n^2 + 10*n - 4 + (6*n^3 - 33*n^2 + 39*n - 12)*x + (10*n^2 - 52*n + 65)*x^2 + (3*n^2 - 3*n - 7)*x^3 + (2*n - 9)*x^4 - x^5)*v[i - 1] + ((2*n^2 - 4*n + 2)*x + (12*n^2 - 19*n + 5)*x^2 + (2*n^2 + 27*n - 36)*x^3 + (11*n + 18)*x^4 + (n + 10)*x^5 + x^6)*deriv(v[i - 1], x))/(-2*n^3 + 10*n^2 - 16*n + 8 + (2*n^3 - 11*n^2 + 28*n - 19)*x + (4*n^2 - 17*n + 13)*x^2 + (n^2 - 1)*x^3 + (n - 1)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-7da9807f7de9f6e4
• Coefficients of one sign. Every nonzero row, n >= 1: 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-7da9807f7de9f6e4
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-7da9807f7de9f6e4
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-7da9807f7de9f6e4

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-7da9807f7de9f6e4

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

  • premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-b176436450506e78 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

    • preserver_image reversal, shift 0 e-73f1cbc14f4fad08

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang_derivative_lag e-b176436450506e78 (expanded above)
        • basis_map A008288 e-796ce5140a0b4b04

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-b8e0a0fa3ba7bf57

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
    • ↵ basis_map A008288 e-796ce5140a0b4b04 (expanded above)
    • ↵ basis_map A008288 e-b8e0a0fa3ba7bf57 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• delannoy → 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.