A008288 → basis A007318

OEIS matches ?

• A209695: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A007482: 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:    5,     5,     1;
   4:   12,    18,     8,     1;
   5:   29,    58,    40,    11,     1;
   6:   70,   175,   164,    71,    14,    1;
   7:  169,   507,   601,   357,   111,   17,    1;
   8:  408,  1428,  2048,  1550,   664,  160,   20,   1;
   9:  985,  3940,  6632,  6106,  3346, 1112,  218,  23,  1;
  10: 2378, 10701, 20680, 22442, 15012, 6394, 1728, 285, 26, 1;

The row polynomials satisfy (n-1)*P(n,x) = (2*n^2-9*n+12-(n-3)*x)*P(n-1,x) - (8*n-28+(8*n-31)*x+(n-5)*x^2)*d/dx P(n-1,x) + (8+16*x+9*x^2+x^3)*d^2/dx^2 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^2 - 9*n + 12 - (n - 3)*x)*P[n - 1, x] - (8*n - 28 + (8*n - 31)*x + (n - 5)*x^2)*D[P[n - 1, x], x] + (8 + 16*x + 9*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
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^2 - 9*n + 12 - (n - 3)*x)*v[i - 1] - (8*n - 28 + (8*n - 31)*x + (n - 5)*x^2)*deriv(v[i - 1], x) + (8 + 16*x + 9*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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)

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,   5;
   4: 1,  8,  18,   12;
   5: 1, 11,  40,   58,   29;
   6: 1, 14,  71,  164,  175,    70;
   7: 1, 17, 111,  357,  601,   507,   169;
   8: 1, 20, 160,  664, 1550,  2048,  1428,   408;
   9: 1, 23, 218, 1112, 3346,  6106,  6632,  3940,   985;
  10: 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378;

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (2*n^2-9*n+12-(n-3)*x)*P(n-1,x) - (8*n-28+(8*n-31)*x+(n-5)*x^2)*d/dx P(n-1,x) + (8+16*x+9*x^2+x^3)*d^2/dx^2 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^2 - 9*n + 12 - (n - 3)*x)*P[n - 1, x] - (8*n - 28 + (8*n - 31)*x + (n - 5)*x^2)*D[P[n - 1, x], x] + (8 + 16*x + 9*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 9*n + 12 - (n - 3)*x)*v[i - 1] - (8*n - 28 + (8*n - 31)*x + (n - 5)*x^2)*deriv(v[i - 1], x) + (8 + 16*x + 9*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); 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, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-84166154a2dbd6d5
• 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-0985ee5bc77b64e0, e-84166154a2dbd6d5
• 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-0985ee5bc77b64e0, e-84166154a2dbd6d5
• 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-0985ee5bc77b64e0, e-84166154a2dbd6d5
• 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-0985ee5bc77b64e0

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-0985ee5bc77b64e0

  Geometric basis map (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 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)
• definition_equality e-6a6707eaf91770cb root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-84166154a2dbd6d5

  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)
    • ↵ preserver_image reversal, shift 0 e-73f1cbc14f4fad08 (expanded above)
    • ↵ basis_map A008288 e-796ce5140a0b4b04 (expanded above)
    • ↵ basis_map A008288 e-b8e0a0fa3ba7bf57 (expanded above)

Related sequences ?

Same fingerprint (3)

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

• delannoy → x ↦ x/(x+1)
• A008288 → x ↦ x/(x+1)
• delannoy → x ↦ x+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