A078812 → basis A007318

OEIS matches ?

• A001353: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A010905: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A195503: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A125662: 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:
   1:    1;
   2:    3,     1;
   3:    8,     6,     1;
   4:   21,    25,     9,     1;
   5:   55,    90,    51,    12,     1;
   6:  144,   300,   234,    86,    15,    1;
   7:  377,   954,   951,   480,   130,   18,    1;
   8:  987,  2939,  3573,  2305,   855,  183,   21,   1;
   9: 2584,  8850, 12707, 10008,  4740, 1386,  245,  24,  1;
  10: 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1;

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

P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (3 + x)*v[i - 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:
  0:    1;
  1:    3,     1;
  2:    8,     6,     1;
  3:   21,    25,     9,     1;
  4:   55,    90,    51,    12,     1;
  5:  144,   300,   234,    86,    15,    1;
  6:  377,   954,   951,   480,   130,   18,    1;
  7:  987,  2939,  3573,  2305,   855,  183,   21,   1;
  8: 2584,  8850, 12707, 10008,  4740, 1386,  245,  24,  1;
  9: 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1;

The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+x)*P(n-1,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[-P[n - 2, x] + (3 + x)*P[n - 1, 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] = -v[i - 2] + (3 + x)*v[i - 1]); 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 >= 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-95c725304ba970c4
• 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-95c725304ba970c4, e-f4c2062c2af62bf3
• 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-95c725304ba970c4, e-f4c2062c2af62bf3
• 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-95c725304ba970c4, e-f4c2062c2af62bf3
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-f4c2062c2af62bf3

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-95c725304ba970c4

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-537ef91063de36e5 root: theorem check

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

    • theorem_check three_term_recurrence e-5aa1c8cf88fab58c root: theorem check

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

• definition_equality e-e9f4e7fbe6961895 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-f4c2062c2af62bf3

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)
    • ↵ theorem_check three_term_recurrence e-5aa1c8cf88fab58c (expanded above)

Related sequences ?

Same fingerprint (3)

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

• A172431 → x ↦ x/(x+1)
• A125662 → reverse
• A125662: A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).

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