A327997: Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).

OEIS matches ?

• A053486: 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 ?
• A327997: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 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:     3,       1;
  2:     9,       7,       1;
  3:    27,      38,      12,       1;
  4:    81,     192,     101,      18,      1;
  5:   243,     969,     755,     215,     25,      1;
  6:   729,    5115,    5494,    2205,    400,     33,     1;
  7:  2187,   29322,   40971,   21469,   5355,    679,    42,    1;
  8:  6561,  187992,  323658,  209356,  66619,  11452,  1078,   52,  1;
  9: 19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1;

The row polynomials satisfy P(n,x) = (n+2+x)*P(n-1,x) - (3*n-3)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(n + 2 + x)*P[n - 1, x] - (3*n - 3)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + 2 + x)*v[i - 1] - (3*n - 3)*v[i - 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)

Triangle begins:
  0:     1;
  1:     3,       1;
  2:     9,       7,       1;
  3:    27,      38,      12,       1;
  4:    81,     192,     101,      18,      1;
  5:   243,     969,     755,     215,     25,      1;
  6:   729,    5115,    5494,    2205,    400,     33,     1;
  7:  2187,   29322,   40971,   21469,   5355,    679,    42,    1;
  8:  6561,  187992,  323658,  209356,  66619,  11452,  1078,   52,  1;
  9: 19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1;

The row polynomials satisfy P(n,x) = (n+2+x)*P(n-1,x) - (3*n-3)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(n + 2 + x)*P[n - 1, x] - (3*n - 3)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + 2 + x)*v[i - 1] - (3*n - 3)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-ea394865db18a818
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-a526bc158f95a9e3, e-ea394865db18a818
• 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-a526bc158f95a9e3
• 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-a526bc158f95a9e3
• 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-a526bc158f95a9e3

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

• theorem_check three_term_recurrence e-a526bc158f95a9e3 root: theorem check

  Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

• theorem_check three_term_recurrence e-ea394865db18a818 root: theorem check

  Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 5 proof steps

• e-5617fd17362674c9 basis_map (A013610) → basis_map_A013610 of Triangle read by rows: coefficients of the polynomials given by Kumm…
• e-8d805b4c99a48239 basis_map (A008277) → basis_map_A008277 of Triangle read by rows: coefficients of the polynomials given by Kumm…
• e-9953a8fe10a45b2a basis_map (A013609) → basis_map_A013609 of Triangle read by rows: coefficients of the polynomials given by Kumm…
• e-cae95ae48665cd8b basis_map (A007318) → basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by Kumm…
• e-e65e9efc7834c6f3 basis_map (A013611) → basis_map_A013611 of Triangle read by rows: coefficients of the polynomials given by Kumm…

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023