A345013: Triangle read by rows, related to clusters of type D.

OEIS matches ?

• A345013: triangle, 36 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A026002: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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,      3;
  2:     15,     20,       6;
  3:     56,    105,      60,      10;
  4:    210,    504,     420,     140,      15;
  5:    792,   2310,    2520,    1260,     280,     21;
  6:   3003,  10296,   13860,    9240,    3150,    504,     28;
  7:  11440,  45045,   72072,   60060,   27720,   6930,    840,    36;
  8:  43758, 194480,  360360,  360360,  210210,  72072,  13860,  1320,   45;
  9: 167960, 831402, 1750320, 2042040, 1441440, 630630, 168168, 25740, 1980, 55;

The row polynomials satisfy (n^2+2*n)*P(n,x) = (4*n^2+6*n+2+(3*n^2+4*n+2)*x)*P(n-1,x) - ((2*n+2)*x+(2*n+2)*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[((4*n^2 + 6*n + 2 + (3*n^2 + 4*n + 2)*x)*P[n - 1, x] - ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*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] = ((4*n^2 + 6*n + 2 + (3*n^2 + 4*n + 2)*x)*v[i - 1] - ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*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)

Triangle begins:
   1:      1;
   2:      4,      3;
   3:     15,     20,       6;
   4:     56,    105,      60,      10;
   5:    210,    504,     420,     140,      15;
   6:    792,   2310,    2520,    1260,     280,     21;
   7:   3003,  10296,   13860,    9240,    3150,    504,     28;
   8:  11440,  45045,   72072,   60060,   27720,   6930,    840,    36;
   9:  43758, 194480,  360360,  360360,  210210,  72072,  13860,  1320,   45;
  10: 167960, 831402, 1750320, 2042040, 1441440, 630630, 168168, 25740, 1980, 55;

The row polynomials satisfy (n^2-1)*P(n,x) = (4*n^2-2*n+(3*n^2-2*n+1)*x)*P(n-1,x) - (2*n*x+2*n*x^2)*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[((4*n^2 - 2*n + (3*n^2 - 2*n + 1)*x)*P[n - 1, x] - (2*n*x + 2*n*x^2)*D[P[n - 1, x], x])/(n^2 - 1)]];
Table[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] = ((4*n^2 - 2*n + (3*n^2 - 2*n + 1)*x)*v[i - 1] - (2*n*x + 2*n*x^2)*deriv(v[i - 1], x))/(n^2 - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #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 >= 0, 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-2ae8453bca041506, e-6f314a2e7485472a
• 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-2ae8453bca041506, e-6f314a2e7485472a
• 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-2ae8453bca041506, e-6f314a2e7485472a
• 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-2ae8453bca041506, e-6f314a2e7485472a

Proof chains

2 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

• preserver_image affine, shift 0 e-2ae8453bca041506

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-32e7e43d5a9debda root: theorem check

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

• preserver_image affine, shift 0 e-6f314a2e7485472a

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check liu_wang e-32e7e43d5a9debda (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